realOTHERauxiliary(3)

NAME
realOTHERauxiliary

SYNOPSIS

  Functions
integer function ilaslc (M, N, A, LDA)
ILASLC scans a matrix for its last non-zero column.
integer function ilaslr (M, N, A, LDA)
ILASLR scans a matrix for its last non-zero row.
subroutine slabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine slacn2 (N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
subroutine slacon (N, V, X, ISGN, EST, KASE)
SLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
subroutine sladiv (A, B, C, D, P, Q)
SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
subroutine sladiv1 (A, B, C, D, P, Q)
real function sladiv2 (A, B, C, D, R, T)
subroutine slaein (RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO)
SLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
subroutine slaexc (WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK, INFO)
SLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
subroutine slag2 (A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)
SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
subroutine slags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.
subroutine slagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
SLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.
subroutine slagv2 (A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR, SNR)
SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.
subroutine slahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)
SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine slahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
SLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
subroutine slaic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
SLAIC1 applies one step of incremental condition estimation.
subroutine slaln2 (LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX, SCALE, XNORM, INFO)
SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
real function slangt (NORM, N, DL, D, DU)
SLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.
real function slanhs (NORM, N, A, LDA, WORK)
SLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.
real function slansb (NORM, UPLO, N, K, AB, LDAB, WORK)
SLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
real function slansp (NORM, UPLO, N, AP, WORK)
SLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.
real function slantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
SLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.
real function slantp (NORM, UPLO, DIAG, N, AP, WORK)
SLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.
real function slantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
SLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.
subroutine slanv2 (A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN)
SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
subroutine slapll (N, X, INCX, Y, INCY, SSMIN)
SLAPLL measures the linear dependence of two vectors.
subroutine slapmr (FORWRD, M, N, X, LDX, K)
SLAPMR rearranges rows of a matrix as specified by a permutation vector.
subroutine slapmt (FORWRD, M, N, X, LDX, K)
SLAPMT performs a forward or backward permutation of the columns of a matrix.
subroutine slaqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
SLAQP2 computes a QR factorization with column pivoting of the matrix block.
subroutine slaqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
subroutine slaqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
subroutine slaqr1 (N, H, LDH, SR1, SI1, SR2, SI2, V)
SLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
subroutine slaqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
subroutine slaqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
subroutine slaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
subroutine slaqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
SLAQR5 performs a single small-bulge multi-shift QR sweep.
subroutine slaqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
SLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.
subroutine slaqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
SLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.
subroutine slaqtr (LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK, INFO)
SLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.
subroutine slar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
subroutine slar2v (N, X, Y, Z, INCX, C, S, INCC)
SLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
subroutine slarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
SLARF applies an elementary reflector to a general rectangular matrix.
subroutine slarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
SLARFB applies a block reflector or its transpose to a general rectangular matrix.
subroutine slarfg (N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
subroutine slarfgp (N, ALPHA, X, INCX, TAU)
SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
subroutine slarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
SLARFT forms the triangular factor T of a block reflector H = I - vtvH
subroutine slarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
SLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.
subroutine slarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK)
SLARFY
subroutine slargv (N, X, INCX, Y, INCY, C, INCC)
SLARGV generates a vector of plane rotations with real cosines and real sines.
subroutine slarrv (N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)
SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
subroutine slartv (N, X, INCX, Y, INCY, C, S, INCC)
SLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors.
subroutine slaswp (N, A, LDA, K1, K2, IPIV, INCX)
SLASWP performs a series of row interchanges on a general rectangular matrix.
subroutine slatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
SLATBS solves a triangular banded system of equations.
subroutine slatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
SLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
subroutine slatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
SLATPS solves a triangular system of equations with the matrix held in packed storage.
subroutine slatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
SLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
subroutine slauu2 (UPLO, N, A, LDA, INFO)
SLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).
subroutine slauum (UPLO, N, A, LDA, INFO)
SLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).
subroutine srscl (N, SA, SX, INCX)
SRSCL multiplies a vector by the reciprocal of a real scalar.
subroutine stprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
STPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.

Detailed Description

This is the group of real other auxiliary routines

Function Documentation

  integer function ilaslc (integer M, integer N, real, dimension( lda, * ) A, integer LDA)

ILASLC scans a matrix for its last non-zero column.

Purpose:

     ILASLC scans A for its last non-zero column.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.

    N

              N is INTEGER
              The number of columns of the matrix A.

    A

              A is REAL array, dimension (LDA,N)
              The m by n matrix A.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2017

Definition at line 80 of file ilaslc.f.

  integer function ilaslr (integer M, integer N, real, dimension( lda, * ) A, integer LDA)

ILASLR scans a matrix for its last non-zero row.

Purpose:

     ILASLR scans A for its last non-zero row.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.

    N

              N is INTEGER
              The number of columns of the matrix A.

    A

              A is REAL array, dimension (LDA,N)
              The m by n matrix A.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 80 of file ilaslr.f.

  subroutine slabrd (integer M, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( ldx, * ) X, integer LDX, real, dimension( ldy, * ) Y, integer LDY)

SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

     SLABRD reduces the first NB rows and columns of a real general
     m by n matrix A to upper or lower bidiagonal form by an orthogonal
     transformation Q**T * A * P, and returns the matrices X and Y which
     are needed to apply the transformation to the unreduced part of A.
     If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
     bidiagonal form.
     This is an auxiliary routine called by SGEBRD

Parameters
    M

              M is INTEGER
              The number of rows in the matrix A.

    N

              N is INTEGER
              The number of columns in the matrix A.

    NB

              NB is INTEGER
              The number of leading rows and columns of A to be reduced.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the m by n general matrix to be reduced.
              On exit, the first NB rows and columns of the matrix are
              overwritten; the rest of the array is unchanged.
              If m >= n, elements on and below the diagonal in the first NB
                columns, with the array TAUQ, represent the orthogonal
                matrix Q as a product of elementary reflectors; and
                elements above the diagonal in the first NB rows, with the
                array TAUP, represent the orthogonal matrix P as a product
                of elementary reflectors.
              If m < n, elements below the diagonal in the first NB
                columns, with the array TAUQ, represent the orthogonal
                matrix Q as a product of elementary reflectors, and
                elements on and above the diagonal in the first NB rows,
                with the array TAUP, represent the orthogonal matrix P as
                a product of elementary reflectors.
              See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    D

              D is REAL array, dimension (NB)
              The diagonal elements of the first NB rows and columns of
              the reduced matrix.  D(i) = A(i,i).

    E

              E is REAL array, dimension (NB)
              The off-diagonal elements of the first NB rows and columns of
              the reduced matrix.

    TAUQ

              TAUQ is REAL array, dimension (NB)
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Q. See Further Details.

    TAUP

              TAUP is REAL array, dimension (NB)
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix P. See Further Details.

    X

              X is REAL array, dimension (LDX,NB)
              The m-by-nb matrix X required to update the unreduced part
              of A.

    LDX

              LDX is INTEGER
              The leading dimension of the array X. LDX >= max(1,M).

    Y

              Y is REAL array, dimension (LDY,NB)
              The n-by-nb matrix Y required to update the unreduced part
              of A.

    LDY

              LDY is INTEGER
              The leading dimension of the array Y. LDY >= max(1,N).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2017

Further Details:

      The matrices Q and P are represented as products of elementary
      reflectors:
         Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
      Each H(i) and G(i) has the form:
         H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
      where tauq and taup are real scalars, and v and u are real vectors.
      If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
      A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
      A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
      A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
      A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      The elements of the vectors v and u together form the m-by-nb matrix
      V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
      the transformation to the unreduced part of the matrix, using a block
      update of the form:  A := A - V*Y**T - X*U**T.
      The contents of A on exit are illustrated by the following examples
      with nb = 2:
      m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
        (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
        (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
        (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
        (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
        (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
        (  v1  v2  a   a   a  )
      where a denotes an element of the original matrix which is unchanged,
      vi denotes an element of the vector defining H(i), and ui an element
      of the vector defining G(i).

Definition at line 212 of file slabrd.f.

  subroutine slacn2 (integer N, real, dimension( * ) V, real, dimension( * ) X, integer, dimension( * ) ISGN, real EST, integer KASE, integer, dimension( 3 ) ISAVE)

SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

Purpose:

     SLACN2 estimates the 1-norm of a square, real matrix A.
     Reverse communication is used for evaluating matrix-vector products.

Parameters
    N

              N is INTEGER
             The order of the matrix.  N >= 1.

    V

              V is REAL array, dimension (N)
             On the final return, V = A*W,  where  EST = norm(V)/norm(W)
             (W is not returned).

    X

              X is REAL array, dimension (N)
             On an intermediate return, X should be overwritten by
                   A * X,   if KASE=1,
                   A**T * X,  if KASE=2,
             and SLACN2 must be re-called with all the other parameters
             unchanged.

    ISGN

              ISGN is INTEGER array, dimension (N)

    EST

              EST is REAL
             On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
             unchanged from the previous call to SLACN2.
             On exit, EST is an estimate (a lower bound) for norm(A).

    KASE

              KASE is INTEGER
             On the initial call to SLACN2, KASE should be 0.
             On an intermediate return, KASE will be 1 or 2, indicating
             whether X should be overwritten by A * X  or A**T * X.
             On the final return from SLACN2, KASE will again be 0.

    ISAVE

              ISAVE is INTEGER array, dimension (3)
             ISAVE is used to save variables between calls to SLACN2

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      Originally named SONEST, dated March 16, 1988.
      This is a thread safe version of SLACON, which uses the array ISAVE
      in place of a SAVE statement, as follows:
         SLACON     SLACN2
          JUMP     ISAVE(1)
          J        ISAVE(2)
          ITER     ISAVE(3)

Contributors:
    Nick Higham, University of Manchester

References:
    N.J. Higham, 'FORTRAN codes for estimating the one-norm of
      a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

Definition at line 138 of file slacn2.f.

  subroutine slacon (integer N, real, dimension( * ) V, real, dimension( * ) X, integer, dimension( * ) ISGN, real EST, integer KASE)

SLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

Purpose:

     SLACON estimates the 1-norm of a square, real matrix A.
     Reverse communication is used for evaluating matrix-vector products.

Parameters
    N

              N is INTEGER
             The order of the matrix.  N >= 1.

    V

              V is REAL array, dimension (N)
             On the final return, V = A*W,  where  EST = norm(V)/norm(W)
             (W is not returned).

    X

              X is REAL array, dimension (N)
             On an intermediate return, X should be overwritten by
                   A * X,   if KASE=1,
                   A**T * X,  if KASE=2,
             and SLACON must be re-called with all the other parameters
             unchanged.

    ISGN

              ISGN is INTEGER array, dimension (N)

    EST

              EST is REAL
             On entry with KASE = 1 or 2 and JUMP = 3, EST should be
             unchanged from the previous call to SLACON.
             On exit, EST is an estimate (a lower bound) for norm(A).

    KASE

              KASE is INTEGER
             On the initial call to SLACON, KASE should be 0.
             On an intermediate return, KASE will be 1 or 2, indicating
             whether X should be overwritten by A * X  or A**T * X.
             On the final return from SLACON, KASE will again be 0.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Contributors:
    Nick Higham, University of Manchester.
     Originally named SONEST, dated March 16, 1988.

References:
    N.J. Higham, 'FORTRAN codes for estimating the one-norm of
      a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

Definition at line 117 of file slacon.f.

  subroutine sladiv (real A, real B, real C, real D, real P, real Q)

SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.

Purpose:

     SLADIV performs complex division in  real arithmetic
                           a + i*b
                p + i*q = ---------
                           c + i*d
     The algorithm is due to Michael Baudin and Robert L. Smith
     and can be found in the paper
     "A Robust Complex Division in Scilab"

Parameters
    A

              A is REAL

    B

              B is REAL

    C

              C is REAL

    D

              D is REAL
              The scalars a, b, c, and d in the above expression.

    P

              P is REAL

    Q

              Q is REAL
              The scalars p and q in the above expression.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    January 2013

Definition at line 93 of file sladiv.f.

  subroutine sladiv1 (real A, real B, real C, real D, real P, real Q)

Definition at line 180 of file sladiv.f.

  real function sladiv2 (real A, real B, real C, real D, real R, real T)

Definition at line 220 of file sladiv.f.

  subroutine slaein (logical RIGHTV, logical NOINIT, integer N, real, dimension( ldh, * ) H, integer LDH, real WR, real WI, real, dimension( * ) VR, real, dimension( * ) VI, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, real EPS3, real SMLNUM, real BIGNUM, integer INFO)

SLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

Purpose:

     SLAEIN uses inverse iteration to find a right or left eigenvector
     corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg
     matrix H.

Parameters
    RIGHTV

              RIGHTV is LOGICAL
              = .TRUE. : compute right eigenvector;
              = .FALSE.: compute left eigenvector.

    NOINIT

              NOINIT is LOGICAL
              = .TRUE. : no initial vector supplied in (VR,VI).
              = .FALSE.: initial vector supplied in (VR,VI).

    N

              N is INTEGER
              The order of the matrix H.  N >= 0.

    H

              H is REAL array, dimension (LDH,N)
              The upper Hessenberg matrix H.

    LDH

              LDH is INTEGER
              The leading dimension of the array H.  LDH >= max(1,N).

    WR

              WR is REAL

    WI

              WI is REAL
              The real and imaginary parts of the eigenvalue of H whose
              corresponding right or left eigenvector is to be computed.

    VR

              VR is REAL array, dimension (N)

    VI

              VI is REAL array, dimension (N)
              On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
              a real starting vector for inverse iteration using the real
              eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
              must contain the real and imaginary parts of a complex
              starting vector for inverse iteration using the complex
              eigenvalue (WR,WI); otherwise VR and VI need not be set.
              On exit, if WI = 0.0 (real eigenvalue), VR contains the
              computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
              VR and VI contain the real and imaginary parts of the
              computed complex eigenvector. The eigenvector is normalized
              so that the component of largest magnitude has magnitude 1;
              here the magnitude of a complex number (x,y) is taken to be
              |x| + |y|.
              VI is not referenced if WI = 0.0.

    B

              B is REAL array, dimension (LDB,N)

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= N+1.

    WORK

              WORK is REAL array, dimension (N)

    EPS3

              EPS3 is REAL
              A small machine-dependent value which is used to perturb
              close eigenvalues, and to replace zero pivots.

    SMLNUM

              SMLNUM is REAL
              A machine-dependent value close to the underflow threshold.

    BIGNUM

              BIGNUM is REAL
              A machine-dependent value close to the overflow threshold.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              = 1:  inverse iteration did not converge; VR is set to the
                    last iterate, and so is VI if WI.ne.0.0.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 174 of file slaein.f.

  subroutine slaexc (logical WANTQ, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldq, * ) Q, integer LDQ, integer J1, integer N1, integer N2, real, dimension( * ) WORK, integer INFO)

SLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.

Purpose:

     SLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
     an upper quasi-triangular matrix T by an orthogonal similarity
     transformation.
     T must be in Schur canonical form, that is, block upper triangular
     with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
     has its diagonal elemnts equal and its off-diagonal elements of
     opposite sign.

Parameters
    WANTQ

              WANTQ is LOGICAL
              = .TRUE. : accumulate the transformation in the matrix Q;
              = .FALSE.: do not accumulate the transformation.

    N

              N is INTEGER
              The order of the matrix T. N >= 0.

    T

              T is REAL array, dimension (LDT,N)
              On entry, the upper quasi-triangular matrix T, in Schur
              canonical form.
              On exit, the updated matrix T, again in Schur canonical form.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= max(1,N).

    Q

              Q is REAL array, dimension (LDQ,N)
              On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
              On exit, if WANTQ is .TRUE., the updated matrix Q.
              If WANTQ is .FALSE., Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.
              LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.

    J1

              J1 is INTEGER
              The index of the first row of the first block T11.

    N1

              N1 is INTEGER
              The order of the first block T11. N1 = 0, 1 or 2.

    N2

              N2 is INTEGER
              The order of the second block T22. N2 = 0, 1 or 2.

    WORK

              WORK is REAL array, dimension (N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              = 1: the transformed matrix T would be too far from Schur
                   form; the blocks are not swapped and T and Q are
                   unchanged.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 140 of file slaexc.f.

  subroutine slag2 (real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real SAFMIN, real SCALE1, real SCALE2, real WR1, real WR2, real WI)

SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

Purpose:

     SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
     problem  A - w B, with scaling as necessary to avoid over-/underflow.
     The scaling factor "s" results in a modified eigenvalue equation
         s A - w B
     where  s  is a non-negative scaling factor chosen so that  w,  w B,
     and  s A  do not overflow and, if possible, do not underflow, either.

Parameters
    A

              A is REAL array, dimension (LDA, 2)
              On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
              is less than 1/SAFMIN.  Entries less than
              sqrt(SAFMIN)*norm(A) are subject to being treated as zero.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= 2.

    B

              B is REAL array, dimension (LDB, 2)
              On entry, the 2 x 2 upper triangular matrix B.  It is
              assumed that the one-norm of B is less than 1/SAFMIN.  The
              diagonals should be at least sqrt(SAFMIN) times the largest
              element of B (in absolute value); if a diagonal is smaller
              than that, then  +/- sqrt(SAFMIN) will be used instead of
              that diagonal.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= 2.

    SAFMIN

              SAFMIN is REAL
              The smallest positive number s.t. 1/SAFMIN does not
              overflow.  (This should always be SLAMCH('S') -- it is an
              argument in order to avoid having to call SLAMCH frequently.)

    SCALE1

              SCALE1 is REAL
              A scaling factor used to avoid over-/underflow in the
              eigenvalue equation which defines the first eigenvalue.  If
              the eigenvalues are complex, then the eigenvalues are
              ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
              exponent range of the machine), SCALE1=SCALE2, and SCALE1
              will always be positive.  If the eigenvalues are real, then
              the first (real) eigenvalue is  WR1 / SCALE1 , but this may
              overflow or underflow, and in fact, SCALE1 may be zero or
              less than the underflow threshold if the exact eigenvalue
              is sufficiently large.

    SCALE2

              SCALE2 is REAL
              A scaling factor used to avoid over-/underflow in the
              eigenvalue equation which defines the second eigenvalue.  If
              the eigenvalues are complex, then SCALE2=SCALE1.  If the
              eigenvalues are real, then the second (real) eigenvalue is
              WR2 / SCALE2 , but this may overflow or underflow, and in
              fact, SCALE2 may be zero or less than the underflow
              threshold if the exact eigenvalue is sufficiently large.

    WR1

              WR1 is REAL
              If the eigenvalue is real, then WR1 is SCALE1 times the
              eigenvalue closest to the (2,2) element of A B**(-1).  If the
              eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
              part of the eigenvalues.

    WR2

              WR2 is REAL
              If the eigenvalue is real, then WR2 is SCALE2 times the
              other eigenvalue.  If the eigenvalue is complex, then
              WR1=WR2 is SCALE1 times the real part of the eigenvalues.

    WI

              WI is REAL
              If the eigenvalue is real, then WI is zero.  If the
              eigenvalue is complex, then WI is SCALE1 times the imaginary
              part of the eigenvalues.  WI will always be non-negative.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2016

Definition at line 158 of file slag2.f.

  subroutine slags2 (logical UPPER, real A1, real A2, real A3, real B1, real B2, real B3, real CSU, real SNU, real CSV, real SNV, real CSQ, real SNQ)

SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.

Purpose:

     SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
     that if ( UPPER ) then
               U**T *A*Q = U**T *( A1 A2 )*Q = ( x  0  )
                                 ( 0  A3 )     ( x  x  )
     and
               V**T*B*Q = V**T *( B1 B2 )*Q = ( x  0  )
                                ( 0  B3 )     ( x  x  )
     or if ( .NOT.UPPER ) then
               U**T *A*Q = U**T *( A1 0  )*Q = ( x  x  )
                                 ( A2 A3 )     ( 0  x  )
     and
               V**T*B*Q = V**T*( B1 0  )*Q = ( x  x  )
                               ( B2 B3 )     ( 0  x  )
     The rows of the transformed A and B are parallel, where
       U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
           ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )
     Z**T denotes the transpose of Z.

Parameters
    UPPER

              UPPER is LOGICAL
              = .TRUE.: the input matrices A and B are upper triangular.
              = .FALSE.: the input matrices A and B are lower triangular.

    A1

              A1 is REAL

    A2

              A2 is REAL

    A3

              A3 is REAL
              On entry, A1, A2 and A3 are elements of the input 2-by-2
              upper (lower) triangular matrix A.

    B1

              B1 is REAL

    B2

              B2 is REAL

    B3

              B3 is REAL
              On entry, B1, B2 and B3 are elements of the input 2-by-2
              upper (lower) triangular matrix B.

    CSU

              CSU is REAL

    SNU

              SNU is REAL
              The desired orthogonal matrix U.

    CSV

              CSV is REAL

    SNV

              SNV is REAL
              The desired orthogonal matrix V.

    CSQ

              CSQ is REAL

    SNQ

              SNQ is REAL
              The desired orthogonal matrix Q.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 154 of file slags2.f.

  subroutine slagtm (character TRANS, integer N, integer NRHS, real ALPHA, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( ldx, * ) X, integer LDX, real BETA, real, dimension( ldb, * ) B, integer LDB)

SLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.

Purpose:

     SLAGTM performs a matrix-vector product of the form
        B := alpha * A * X + beta * B
     where A is a tridiagonal matrix of order N, B and X are N by NRHS
     matrices, and alpha and beta are real scalars, each of which may be
     0., 1., or -1.

Parameters
    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  No transpose, B := alpha * A * X + beta * B
              = 'T':  Transpose,    B := alpha * A'* X + beta * B
              = 'C':  Conjugate transpose = Transpose

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices X and B.

    ALPHA

              ALPHA is REAL
              The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
              it is assumed to be 0.

    DL

              DL is REAL array, dimension (N-1)
              The (n-1) sub-diagonal elements of T.

    D

              D is REAL array, dimension (N)
              The diagonal elements of T.

    DU

              DU is REAL array, dimension (N-1)
              The (n-1) super-diagonal elements of T.

    X

              X is REAL array, dimension (LDX,NRHS)
              The N by NRHS matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.  LDX >= max(N,1).

    BETA

              BETA is REAL
              The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
              it is assumed to be 1.

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the N by NRHS matrix B.
              On exit, B is overwritten by the matrix expression
              B := alpha * A * X + beta * B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(N,1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 147 of file slagtm.f.

  subroutine slagv2 (real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( 2 ) ALPHAR, real, dimension( 2 ) ALPHAI, real, dimension( 2 ) BETA, real CSL, real SNL, real CSR, real SNR)

SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.

Purpose:

     SLAGV2 computes the Generalized Schur factorization of a real 2-by-2
     matrix pencil (A,B) where B is upper triangular. This routine
     computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
     SNR such that
     1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
        types), then
        [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
        [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
        [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
        [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],
     2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
        then
        [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
        [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
        [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
        [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]
        where b11 >= b22 > 0.

Parameters
    A

              A is REAL array, dimension (LDA, 2)
              On entry, the 2 x 2 matrix A.
              On exit, A is overwritten by the ``A-part'' of the
              generalized Schur form.

    LDA

              LDA is INTEGER
              THe leading dimension of the array A.  LDA >= 2.

    B

              B is REAL array, dimension (LDB, 2)
              On entry, the upper triangular 2 x 2 matrix B.
              On exit, B is overwritten by the ``B-part'' of the
              generalized Schur form.

    LDB

              LDB is INTEGER
              THe leading dimension of the array B.  LDB >= 2.

    ALPHAR

              ALPHAR is REAL array, dimension (2)

    ALPHAI

              ALPHAI is REAL array, dimension (2)

    BETA

              BETA is REAL array, dimension (2)
              (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
              pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may
              be zero.

    CSL

              CSL is REAL
              The cosine of the left rotation matrix.

    SNL

              SNL is REAL
              The sine of the left rotation matrix.

    CSR

              CSR is REAL
              The cosine of the right rotation matrix.

    SNR

              SNR is REAL
              The sine of the right rotation matrix.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Contributors:
    Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 159 of file slagv2.f.

  subroutine slahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, integer INFO)

SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

        SLAHQR is an auxiliary routine called by SHSEQR to update the
        eigenvalues and Schur decomposition already computed by SHSEQR, by
        dealing with the Hessenberg submatrix in rows and columns ILO to
        IHI.

Parameters
    WANTT

              WANTT is LOGICAL
              = .TRUE. : the full Schur form T is required;
              = .FALSE.: only eigenvalues are required.

    WANTZ

              WANTZ is LOGICAL
              = .TRUE. : the matrix of Schur vectors Z is required;
              = .FALSE.: Schur vectors are not required.

    N

              N is INTEGER
              The order of the matrix H.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              It is assumed that H is already upper quasi-triangular in
              rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
              ILO = 1). SLAHQR works primarily with the Hessenberg
              submatrix in rows and columns ILO to IHI, but applies
              transformations to all of H if WANTT is .TRUE..
              1 <= ILO <= max(1,IHI); IHI <= N.

    H

              H is REAL array, dimension (LDH,N)
              On entry, the upper Hessenberg matrix H.
              On exit, if INFO is zero and if WANTT is .TRUE., H is upper
              quasi-triangular in rows and columns ILO:IHI, with any
              2-by-2 diagonal blocks in standard form. If INFO is zero
              and WANTT is .FALSE., the contents of H are unspecified on
              exit.  The output state of H if INFO is nonzero is given
              below under the description of INFO.

    LDH

              LDH is INTEGER
              The leading dimension of the array H. LDH >= max(1,N).

    WR

              WR is REAL array, dimension (N)

    WI

              WI is REAL array, dimension (N)
              The real and imaginary parts, respectively, of the computed
              eigenvalues ILO to IHI are stored in the corresponding
              elements of WR and WI. If two eigenvalues are computed as a
              complex conjugate pair, they are stored in consecutive
              elements of WR and WI, say the i-th and (i+1)th, with
              WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
              eigenvalues are stored in the same order as on the diagonal
              of the Schur form returned in H, with WR(i) = H(i,i), and, if
              H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
              WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

    ILOZ

              ILOZ is INTEGER

    IHIZ

              IHIZ is INTEGER
              Specify the rows of Z to which transformations must be
              applied if WANTZ is .TRUE..
              1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

    Z

              Z is REAL array, dimension (LDZ,N)
              If WANTZ is .TRUE., on entry Z must contain the current
              matrix Z of transformations accumulated by SHSEQR, and on
              exit Z has been updated; transformations are applied only to
              the submatrix Z(ILOZ:IHIZ,ILO:IHI).
              If WANTZ is .FALSE., Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z. LDZ >= max(1,N).

    INFO

              INFO is INTEGER
               = 0:   successful exit
               > 0:   If INFO = i, SLAHQR failed to compute all the
                      eigenvalues ILO to IHI in a total of 30 iterations
                      per eigenvalue; elements i+1:ihi of WR and WI
                      contain those eigenvalues which have been
                      successfully computed.
                      If INFO > 0 and WANTT is .FALSE., then on exit,
                      the remaining unconverged eigenvalues are the
                      eigenvalues of the upper Hessenberg matrix rows
                      and columns ILO through INFO of the final, output
                      value of H.
                      If INFO > 0 and WANTT is .TRUE., then on exit
              (*)       (initial value of H)*U  = U*(final value of H)
                      where U is an orthogonal matrix.    The final
                      value of H is upper Hessenberg and triangular in
                      rows and columns INFO+1 through IHI.
                      If INFO > 0 and WANTZ is .TRUE., then on exit
                          (final value of Z)  = (initial value of Z)*U
                      where U is the orthogonal matrix in (*)
                      (regardless of the value of WANTT.)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

         02-96 Based on modifications by
         David Day, Sandia National Laboratory, USA
         12-04 Further modifications by
         Ralph Byers, University of Kansas, USA
         This is a modified version of SLAHQR from LAPACK version 3.0.
         It is (1) more robust against overflow and underflow and
         (2) adopts the more conservative Ahues & Tisseur stopping
         criterion (LAWN 122, 1997).

Definition at line 209 of file slahqr.f.

  subroutine slahr2 (integer N, integer K, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( nb ) TAU, real, dimension( ldt, nb ) T, integer LDT, real, dimension( ldy, nb ) Y, integer LDY)

SLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Purpose:

     SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
     matrix A so that elements below the k-th subdiagonal are zero. The
     reduction is performed by an orthogonal similarity transformation
     Q**T * A * Q. The routine returns the matrices V and T which determine
     Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
     This is an auxiliary routine called by SGEHRD.

Parameters
    N

              N is INTEGER
              The order of the matrix A.

    K

              K is INTEGER
              The offset for the reduction. Elements below the k-th
              subdiagonal in the first NB columns are reduced to zero.
              K < N.

    NB

              NB is INTEGER
              The number of columns to be reduced.

    A

              A is REAL array, dimension (LDA,N-K+1)
              On entry, the n-by-(n-k+1) general matrix A.
              On exit, the elements on and above the k-th subdiagonal in
              the first NB columns are overwritten with the corresponding
              elements of the reduced matrix; the elements below the k-th
              subdiagonal, with the array TAU, represent the matrix Q as a
              product of elementary reflectors. The other columns of A are
              unchanged. See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    TAU

              TAU is REAL array, dimension (NB)
              The scalar factors of the elementary reflectors. See Further
              Details.

    T

              T is REAL array, dimension (LDT,NB)
              The upper triangular matrix T.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= NB.

    Y

              Y is REAL array, dimension (LDY,NB)
              The n-by-nb matrix Y.

    LDY

              LDY is INTEGER
              The leading dimension of the array Y. LDY >= N.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      The matrix Q is represented as a product of nb elementary reflectors
         Q = H(1) H(2) . . . H(nb).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
      A(i+k+1:n,i), and tau in TAU(i).
      The elements of the vectors v together form the (n-k+1)-by-nb matrix
      V which is needed, with T and Y, to apply the transformation to the
      unreduced part of the matrix, using an update of the form:
      A := (I - V*T*V**T) * (A - Y*V**T).
      The contents of A on exit are illustrated by the following example
      with n = 7, k = 3 and nb = 2:
         ( a   a   a   a   a )
         ( a   a   a   a   a )
         ( a   a   a   a   a )
         ( h   h   a   a   a )
         ( v1  h   a   a   a )
         ( v1  v2  a   a   a )
         ( v1  v2  a   a   a )
      where a denotes an element of the original matrix A, h denotes a
      modified element of the upper Hessenberg matrix H, and vi denotes an
      element of the vector defining H(i).
      This subroutine is a slight modification of LAPACK-3.0's DLAHRD
      incorporating improvements proposed by Quintana-Orti and Van de
      Gejin. Note that the entries of A(1:K,2:NB) differ from those
      returned by the original LAPACK-3.0's DLAHRD routine. (This
      subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)

References:
    Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
      performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

Definition at line 183 of file slahr2.f.

  subroutine slaic1 (integer JOB, integer J, real, dimension( j ) X, real SEST, real, dimension( j ) W, real GAMMA, real SESTPR, real S, real C)

SLAIC1 applies one step of incremental condition estimation.

Purpose:

     SLAIC1 applies one step of incremental condition estimation in
     its simplest version:
     Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
     lower triangular matrix L, such that
              twonorm(L*x) = sest
     Then SLAIC1 computes sestpr, s, c such that
     the vector
                     [ s*x ]
              xhat = [  c  ]
     is an approximate singular vector of
                     [ L      0  ]
              Lhat = [ w**T gamma ]
     in the sense that
              twonorm(Lhat*xhat) = sestpr.
     Depending on JOB, an estimate for the largest or smallest singular
     value is computed.
     Note that [s c]**T and sestpr**2 is an eigenpair of the system
         diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
                                               [ gamma ]
     where  alpha =  x**T*w.

Parameters
    JOB

              JOB is INTEGER
              = 1: an estimate for the largest singular value is computed.
              = 2: an estimate for the smallest singular value is computed.

    J

              J is INTEGER
              Length of X and W

    X

              X is REAL array, dimension (J)
              The j-vector x.

    SEST

              SEST is REAL
              Estimated singular value of j by j matrix L

    W

              W is REAL array, dimension (J)
              The j-vector w.

    GAMMA

              GAMMA is REAL
              The diagonal element gamma.

    SESTPR

              SESTPR is REAL
              Estimated singular value of (j+1) by (j+1) matrix Lhat.

    S

              S is REAL
              Sine needed in forming xhat.

    C

              C is REAL
              Cosine needed in forming xhat.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 136 of file slaic1.f.

  subroutine slaln2 (logical LTRANS, integer NA, integer NW, real SMIN, real CA, real, dimension( lda, * ) A, integer LDA, real D1, real D2, real, dimension( ldb, * ) B, integer LDB, real WR, real WI, real, dimension( ldx, * ) X, integer LDX, real SCALE, real XNORM, integer INFO)

SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

Purpose:

     SLALN2 solves a system of the form  (ca A - w D ) X = s B
     or (ca A**T - w D) X = s B   with possible scaling ("s") and
     perturbation of A.  (A**T means A-transpose.)
     A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
     real diagonal matrix, w is a real or complex value, and X and B are
     NA x 1 matrices -- real if w is real, complex if w is complex.  NA
     may be 1 or 2.
     If w is complex, X and B are represented as NA x 2 matrices,
     the first column of each being the real part and the second
     being the imaginary part.
     "s" is a scaling factor (<= 1), computed by SLALN2, which is
     so chosen that X can be computed without overflow.  X is further
     scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
     than overflow.
     If both singular values of (ca A - w D) are less than SMIN,
     SMIN*identity will be used instead of (ca A - w D).  If only one
     singular value is less than SMIN, one element of (ca A - w D) will be
     perturbed enough to make the smallest singular value roughly SMIN.
     If both singular values are at least SMIN, (ca A - w D) will not be
     perturbed.  In any case, the perturbation will be at most some small
     multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
     are computed by infinity-norm approximations, and thus will only be
     correct to a factor of 2 or so.
     Note: all input quantities are assumed to be smaller than overflow
     by a reasonable factor.  (See BIGNUM.)

Parameters
    LTRANS

              LTRANS is LOGICAL
              =.TRUE.:  A-transpose will be used.
              =.FALSE.: A will be used (not transposed.)

    NA

              NA is INTEGER
              The size of the matrix A.  It may (only) be 1 or 2.

    NW

              NW is INTEGER
              1 if "w" is real, 2 if "w" is complex.  It may only be 1
              or 2.

    SMIN

              SMIN is REAL
              The desired lower bound on the singular values of A.  This
              should be a safe distance away from underflow or overflow,
              say, between (underflow/machine precision) and  (machine
              precision * overflow ).  (See BIGNUM and ULP.)

    CA

              CA is REAL
              The coefficient c, which A is multiplied by.

    A

              A is REAL array, dimension (LDA,NA)
              The NA x NA matrix A.

    LDA

              LDA is INTEGER
              The leading dimension of A.  It must be at least NA.

    D1

              D1 is REAL
              The 1,1 element in the diagonal matrix D.

    D2

              D2 is REAL
              The 2,2 element in the diagonal matrix D.  Not used if NA=1.

    B

              B is REAL array, dimension (LDB,NW)
              The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
              complex), column 1 contains the real part of B and column 2
              contains the imaginary part.

    LDB

              LDB is INTEGER
              The leading dimension of B.  It must be at least NA.

    WR

              WR is REAL
              The real part of the scalar "w".

    WI

              WI is REAL
              The imaginary part of the scalar "w".  Not used if NW=1.

    X

              X is REAL array, dimension (LDX,NW)
              The NA x NW matrix X (unknowns), as computed by SLALN2.
              If NW=2 ("w" is complex), on exit, column 1 will contain
              the real part of X and column 2 will contain the imaginary
              part.

    LDX

              LDX is INTEGER
              The leading dimension of X.  It must be at least NA.

    SCALE

              SCALE is REAL
              The scale factor that B must be multiplied by to insure
              that overflow does not occur when computing X.  Thus,
              (ca A - w D) X  will be SCALE*B, not B (ignoring
              perturbations of A.)  It will be at most 1.

    XNORM

              XNORM is REAL
              The infinity-norm of X, when X is regarded as an NA x NW
              real matrix.

    INFO

              INFO is INTEGER
              An error flag.  It will be set to zero if no error occurs,
              a negative number if an argument is in error, or a positive
              number if  ca A - w D  had to be perturbed.
              The possible values are:
              = 0: No error occurred, and (ca A - w D) did not have to be
                     perturbed.
              = 1: (ca A - w D) had to be perturbed to make its smallest
                   (or only) singular value greater than SMIN.
              NOTE: In the interests of speed, this routine does not
                    check the inputs for errors.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 220 of file slaln2.f.

  real function slangt (character NORM, integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU)

SLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.

Purpose:

     SLANGT  returns the value of the one norm,  or the Frobenius norm, or
     the  infinity norm,  or the  element of  largest absolute value  of a
     real tridiagonal matrix A.

Returns
    SLANGT

        SLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in SLANGT as described
              above.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.  When N = 0, SLANGT is
              set to zero.

    DL

              DL is REAL array, dimension (N-1)
              The (n-1) sub-diagonal elements of A.

    D

              D is REAL array, dimension (N)
              The diagonal elements of A.

    DU

              DU is REAL array, dimension (N-1)
              The (n-1) super-diagonal elements of A.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 108 of file slangt.f.

  real function slanhs (character NORM, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)

SLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.

Purpose:

     SLANHS  returns the value of the one norm,  or the Frobenius norm, or
     the  infinity norm,  or the  element of  largest absolute value  of a
     Hessenberg matrix A.

Returns
    SLANHS

        SLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in SLANHS as described
              above.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.  When N = 0, SLANHS is
              set to zero.

    A

              A is REAL array, dimension (LDA,N)
              The n by n upper Hessenberg matrix A; the part of A below the
              first sub-diagonal is not referenced.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(N,1).

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK)),
              where LWORK >= N when NORM = 'I'; otherwise, WORK is not
              referenced.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 110 of file slanhs.f.

  real function slansb (character NORM, character UPLO, integer N, integer K, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)

SLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.

Purpose:

     SLANSB  returns the value of the one norm,  or the Frobenius norm, or
     the  infinity norm,  or the element of  largest absolute value  of an
     n by n symmetric band matrix A,  with k super-diagonals.

Returns
    SLANSB

        SLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in SLANSB as described
              above.

    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              band matrix A is supplied.
              = 'U':  Upper triangular part is supplied
              = 'L':  Lower triangular part is supplied

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.  When N = 0, SLANSB is
              set to zero.

    K

              K is INTEGER
              The number of super-diagonals or sub-diagonals of the
              band matrix A.  K >= 0.

    AB

              AB is REAL array, dimension (LDAB,N)
              The upper or lower triangle of the symmetric band matrix A,
              stored in the first K+1 rows of AB.  The j-th column of A is
              stored in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= K+1.

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK)),
              where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
              WORK is not referenced.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 131 of file slansb.f.

  real function slansp (character NORM, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) WORK)

SLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.

Purpose:

     SLANSP  returns the value of the one norm,  or the Frobenius norm, or
     the  infinity norm,  or the  element of  largest absolute value  of a
     real symmetric matrix A,  supplied in packed form.

Returns
    SLANSP

        SLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in SLANSP as described
              above.

    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is supplied.
              = 'U':  Upper triangular part of A is supplied
              = 'L':  Lower triangular part of A is supplied

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.  When N = 0, SLANSP is
              set to zero.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              The upper or lower triangle of the symmetric matrix A, packed
              columnwise in a linear array.  The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK)),
              where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
              WORK is not referenced.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 116 of file slansp.f.

  real function slantb (character NORM, character UPLO, character DIAG, integer N, integer K, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)

SLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.

Purpose:

     SLANTB  returns the value of the one norm,  or the Frobenius norm, or
     the  infinity norm,  or the element of  largest absolute value  of an
     n by n triangular band matrix A,  with ( k + 1 ) diagonals.

Returns
    SLANTB

        SLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in SLANTB as described
              above.

    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.  When N = 0, SLANTB is
              set to zero.

    K

              K is INTEGER
              The number of super-diagonals of the matrix A if UPLO = 'U',
              or the number of sub-diagonals of the matrix A if UPLO = 'L'.
              K >= 0.

    AB

              AB is REAL array, dimension (LDAB,N)
              The upper or lower triangular band matrix A, stored in the
              first k+1 rows of AB.  The j-th column of A is stored
              in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
              Note that when DIAG = 'U', the elements of the array AB
              corresponding to the diagonal elements of the matrix A are
              not referenced, but are assumed to be one.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= K+1.

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK)),
              where LWORK >= N when NORM = 'I'; otherwise, WORK is not
              referenced.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 142 of file slantb.f.

  real function slantp (character NORM, character UPLO, character DIAG, integer N, real, dimension( * ) AP, real, dimension( * ) WORK)

SLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.

Purpose:

     SLANTP  returns the value of the one norm,  or the Frobenius norm, or
     the  infinity norm,  or the  element of  largest absolute value  of a
     triangular matrix A, supplied in packed form.

Returns
    SLANTP

        SLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in SLANTP as described
              above.

    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.  When N = 0, SLANTP is
              set to zero.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              The upper or lower triangular matrix A, packed columnwise in
              a linear array.  The j-th column of A is stored in the array
              AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              Note that when DIAG = 'U', the elements of the array AP
              corresponding to the diagonal elements of the matrix A are
              not referenced, but are assumed to be one.

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK)),
              where LWORK >= N when NORM = 'I'; otherwise, WORK is not
              referenced.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 126 of file slantp.f.

  real function slantr (character NORM, character UPLO, character DIAG, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)

SLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.

Purpose:

     SLANTR  returns the value of the one norm,  or the Frobenius norm, or
     the  infinity norm,  or the  element of  largest absolute value  of a
     trapezoidal or triangular matrix A.

Returns
    SLANTR

        SLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in SLANTR as described
              above.

    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower trapezoidal.
              = 'U':  Upper trapezoidal
              = 'L':  Lower trapezoidal
              Note that A is triangular instead of trapezoidal if M = N.

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A has unit diagonal.
              = 'N':  Non-unit diagonal
              = 'U':  Unit diagonal

    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0, and if
              UPLO = 'U', M <= N.  When M = 0, SLANTR is set to zero.

    N

              N is INTEGER
              The number of columns of the matrix A.  N >= 0, and if
              UPLO = 'L', N <= M.  When N = 0, SLANTR is set to zero.

    A

              A is REAL array, dimension (LDA,N)
              The trapezoidal matrix A (A is triangular if M = N).
              If UPLO = 'U', the leading m by n upper trapezoidal part of
              the array A contains the upper trapezoidal matrix, and the
              strictly lower triangular part of A is not referenced.
              If UPLO = 'L', the leading m by n lower trapezoidal part of
              the array A contains the lower trapezoidal matrix, and the
              strictly upper triangular part of A is not referenced.  Note
              that when DIAG = 'U', the diagonal elements of A are not
              referenced and are assumed to be one.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(M,1).

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK)),
              where LWORK >= M when NORM = 'I'; otherwise, WORK is not
              referenced.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 143 of file slantr.f.

  subroutine slanv2 (real A, real B, real C, real D, real RT1R, real RT1I, real RT2R, real RT2I, real CS, real SN)

SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.

Purpose:

     SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
     matrix in standard form:
          [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
          [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]
     where either
     1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
     2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
     conjugate eigenvalues.

Parameters
    A

              A is REAL

    B

              B is REAL

    C

              C is REAL

    D

              D is REAL
              On entry, the elements of the input matrix.
              On exit, they are overwritten by the elements of the
              standardised Schur form.

    RT1R

              RT1R is REAL

    RT1I

              RT1I is REAL

    RT2R

              RT2R is REAL

    RT2I

              RT2I is REAL
              The real and imaginary parts of the eigenvalues. If the
              eigenvalues are a complex conjugate pair, RT1I > 0.

    CS

              CS is REAL

    SN

              SN is REAL
              Parameters of the rotation matrix.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      Modified by V. Sima, Research Institute for Informatics, Bucharest,
      Romania, to reduce the risk of cancellation errors,
      when computing real eigenvalues, and to ensure, if possible, that
      abs(RT1R) >= abs(RT2R).

Definition at line 129 of file slanv2.f.

  subroutine slapll (integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y, integer INCY, real SSMIN)

SLAPLL measures the linear dependence of two vectors.

Purpose:

     Given two column vectors X and Y, let
                          A = ( X Y ).
     The subroutine first computes the QR factorization of A = Q*R,
     and then computes the SVD of the 2-by-2 upper triangular matrix R.
     The smaller singular value of R is returned in SSMIN, which is used
     as the measurement of the linear dependency of the vectors X and Y.

Parameters
    N

              N is INTEGER
              The length of the vectors X and Y.

    X

              X is REAL array,
                             dimension (1+(N-1)*INCX)
              On entry, X contains the N-vector X.
              On exit, X is overwritten.

    INCX

              INCX is INTEGER
              The increment between successive elements of X. INCX > 0.

    Y

              Y is REAL array,
                             dimension (1+(N-1)*INCY)
              On entry, Y contains the N-vector Y.
              On exit, Y is overwritten.

    INCY

              INCY is INTEGER
              The increment between successive elements of Y. INCY > 0.

    SSMIN

              SSMIN is REAL
              The smallest singular value of the N-by-2 matrix A = ( X Y ).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 104 of file slapll.f.

  subroutine slapmr (logical FORWRD, integer M, integer N, real, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)

SLAPMR rearranges rows of a matrix as specified by a permutation vector.

Purpose:

     SLAPMR rearranges the rows of the M by N matrix X as specified
     by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
     If FORWRD = .TRUE.,  forward permutation:
          X(K(I),*) is moved X(I,*) for I = 1,2,...,M.
     If FORWRD = .FALSE., backward permutation:
          X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.

Parameters
    FORWRD

              FORWRD is LOGICAL
              = .TRUE., forward permutation
              = .FALSE., backward permutation

    M

              M is INTEGER
              The number of rows of the matrix X. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix X. N >= 0.

    X

              X is REAL array, dimension (LDX,N)
              On entry, the M by N matrix X.
              On exit, X contains the permuted matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X, LDX >= MAX(1,M).

    K

              K is INTEGER array, dimension (M)
              On entry, K contains the permutation vector. K is used as
              internal workspace, but reset to its original value on
              output.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 106 of file slapmr.f.

  subroutine slapmt (logical FORWRD, integer M, integer N, real, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)

SLAPMT performs a forward or backward permutation of the columns of a matrix.

Purpose:

     SLAPMT rearranges the columns of the M by N matrix X as specified
     by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
     If FORWRD = .TRUE.,  forward permutation:
          X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.
     If FORWRD = .FALSE., backward permutation:
          X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.

Parameters
    FORWRD

              FORWRD is LOGICAL
              = .TRUE., forward permutation
              = .FALSE., backward permutation

    M

              M is INTEGER
              The number of rows of the matrix X. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix X. N >= 0.

    X

              X is REAL array, dimension (LDX,N)
              On entry, the M by N matrix X.
              On exit, X contains the permuted matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X, LDX >= MAX(1,M).

    K

              K is INTEGER array, dimension (N)
              On entry, K contains the permutation vector. K is used as
              internal workspace, but reset to its original value on
              output.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 106 of file slapmt.f.

  subroutine slaqp2 (integer M, integer N, integer OFFSET, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, real, dimension( * ) WORK)

SLAQP2 computes a QR factorization with column pivoting of the matrix block.

Purpose:

     SLAQP2 computes a QR factorization with column pivoting of
     the block A(OFFSET+1:M,1:N).
     The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. N >= 0.

    OFFSET

              OFFSET is INTEGER
              The number of rows of the matrix A that must be pivoted
              but no factorized. OFFSET >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
              the triangular factor obtained; the elements in block
              A(OFFSET+1:M,1:N) below the diagonal, together with the
              array TAU, represent the orthogonal matrix Q as a product of
              elementary reflectors. Block A(1:OFFSET,1:N) has been
              accordingly pivoted, but no factorized.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    JPVT

              JPVT is INTEGER array, dimension (N)
              On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
              to the front of A*P (a leading column); if JPVT(i) = 0,
              the i-th column of A is a free column.
              On exit, if JPVT(i) = k, then the i-th column of A*P
              was the k-th column of A.

    TAU

              TAU is REAL array, dimension (min(M,N))
              The scalar factors of the elementary reflectors.

    VN1

              VN1 is REAL array, dimension (N)
              The vector with the partial column norms.

    VN2

              VN2 is REAL array, dimension (N)
              The vector with the exact column norms.

    WORK

              WORK is REAL array, dimension (N)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Contributors:
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA
     Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:
    LAPACK Working Note 176

Definition at line 151 of file slaqp2.f.

  subroutine slaqps (integer M, integer N, integer OFFSET, integer NB, integer KB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, real, dimension( * ) AUXV, real, dimension( ldf, * ) F, integer LDF)

SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.

Purpose:

     SLAQPS computes a step of QR factorization with column pivoting
     of a real M-by-N matrix A by using Blas-3.  It tries to factorize
     NB columns from A starting from the row OFFSET+1, and updates all
     of the matrix with Blas-3 xGEMM.
     In some cases, due to catastrophic cancellations, it cannot
     factorize NB columns.  Hence, the actual number of factorized
     columns is returned in KB.
     Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. N >= 0

    OFFSET

              OFFSET is INTEGER
              The number of rows of A that have been factorized in
              previous steps.

    NB

              NB is INTEGER
              The number of columns to factorize.

    KB

              KB is INTEGER
              The number of columns actually factorized.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, block A(OFFSET+1:M,1:KB) is the triangular
              factor obtained and block A(1:OFFSET,1:N) has been
              accordingly pivoted, but no factorized.
              The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
              been updated.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    JPVT

              JPVT is INTEGER array, dimension (N)
              JPVT(I) = K <==> Column K of the full matrix A has been
              permuted into position I in AP.

    TAU

              TAU is REAL array, dimension (KB)
              The scalar factors of the elementary reflectors.

    VN1

              VN1 is REAL array, dimension (N)
              The vector with the partial column norms.

    VN2

              VN2 is REAL array, dimension (N)
              The vector with the exact column norms.

    AUXV

              AUXV is REAL array, dimension (NB)
              Auxiliary vector.

    F

              F is REAL array, dimension (LDF,NB)
              Matrix F**T = L*Y**T*A.

    LDF

              LDF is INTEGER
              The leading dimension of the array F. LDF >= max(1,N).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Contributors:
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA

 Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:
    LAPACK Working Note 176

Definition at line 180 of file slaqps.f.

  subroutine slaqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)

SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

        SLAQR0 computes the eigenvalues of a Hessenberg matrix H
        and, optionally, the matrices T and Z from the Schur decomposition
        H = Z T Z**T, where T is an upper quasi-triangular matrix (the
        Schur form), and Z is the orthogonal matrix of Schur vectors.
        Optionally Z may be postmultiplied into an input orthogonal
        matrix Q so that this routine can give the Schur factorization
        of a matrix A which has been reduced to the Hessenberg form H
        by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

Parameters
    WANTT

              WANTT is LOGICAL
              = .TRUE. : the full Schur form T is required;
              = .FALSE.: only eigenvalues are required.

    WANTZ

              WANTZ is LOGICAL
              = .TRUE. : the matrix of Schur vectors Z is required;
              = .FALSE.: Schur vectors are not required.

    N

              N is INTEGER
               The order of the matrix H.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
               It is assumed that H is already upper triangular in rows
               and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
               H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
               previous call to SGEBAL, and then passed to SGEHRD when the
               matrix output by SGEBAL is reduced to Hessenberg form.
               Otherwise, ILO and IHI should be set to 1 and N,
               respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
               If N = 0, then ILO = 1 and IHI = 0.

    H

              H is REAL array, dimension (LDH,N)
               On entry, the upper Hessenberg matrix H.
               On exit, if INFO = 0 and WANTT is .TRUE., then H contains
               the upper quasi-triangular matrix T from the Schur
               decomposition (the Schur form); 2-by-2 diagonal blocks
               (corresponding to complex conjugate pairs of eigenvalues)
               are returned in standard form, with H(i,i) = H(i+1,i+1)
               and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
               .FALSE., then the contents of H are unspecified on exit.
               (The output value of H when INFO > 0 is given under the
               description of INFO below.)
               This subroutine may explicitly set H(i,j) = 0 for i > j and
               j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

    LDH

              LDH is INTEGER
               The leading dimension of the array H. LDH >= max(1,N).

    WR

              WR is REAL array, dimension (IHI)

    WI

              WI is REAL array, dimension (IHI)
               The real and imaginary parts, respectively, of the computed
               eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
               and WI(ILO:IHI). If two eigenvalues are computed as a
               complex conjugate pair, they are stored in consecutive
               elements of WR and WI, say the i-th and (i+1)th, with
               WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
               the eigenvalues are stored in the same order as on the
               diagonal of the Schur form returned in H, with
               WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
               block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
               WI(i+1) = -WI(i).

    ILOZ

              ILOZ is INTEGER

    IHIZ

              IHIZ is INTEGER
               Specify the rows of Z to which transformations must be
               applied if WANTZ is .TRUE..
               1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

    Z

              Z is REAL array, dimension (LDZ,IHI)
               If WANTZ is .FALSE., then Z is not referenced.
               If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
               replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
               orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
               (The output value of Z when INFO > 0 is given under
               the description of INFO below.)

    LDZ

              LDZ is INTEGER
               The leading dimension of the array Z.  if WANTZ is .TRUE.
               then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

    WORK

              WORK is REAL array, dimension LWORK
               On exit, if LWORK = -1, WORK(1) returns an estimate of
               the optimal value for LWORK.

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK.  LWORK >= max(1,N)
               is sufficient, but LWORK typically as large as 6*N may
               be required for optimal performance.  A workspace query
               to determine the optimal workspace size is recommended.
               If LWORK = -1, then SLAQR0 does a workspace query.
               In this case, SLAQR0 checks the input parameters and
               estimates the optimal workspace size for the given
               values of N, ILO and IHI.  The estimate is returned
               in WORK(1).  No error message related to LWORK is
               issued by XERBLA.  Neither H nor Z are accessed.

    INFO

              INFO is INTEGER
                 = 0:  successful exit
                 > 0:  if INFO = i, SLAQR0 failed to compute all of
                    the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                    and WI contain those eigenvalues which have been
                    successfully computed.  (Failures are rare.)
                    If INFO > 0 and WANT is .FALSE., then on exit,
                    the remaining unconverged eigenvalues are the eigen-
                    values of the upper Hessenberg matrix rows and
                    columns ILO through INFO of the final, output
                    value of H.
                    If INFO > 0 and WANTT is .TRUE., then on exit
               (*)  (initial value of H)*U  = U*(final value of H)
                    where U is an orthogonal matrix.  The final
                    value of H is upper Hessenberg and quasi-triangular
                    in rows and columns INFO+1 through IHI.
                    If INFO > 0 and WANTZ is .TRUE., then on exit
                      (final value of Z(ILO:IHI,ILOZ:IHIZ)
                       =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
                    where U is the orthogonal matrix in (*) (regard-
                    less of the value of WANTT.)
                    If INFO > 0 and WANTZ is .FALSE., then Z is not
                    accessed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Contributors:
    Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

      K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
      Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
      Performance, SIAM Journal of Matrix Analysis, volume 23, pages
      929--947, 2002.

     K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 258 of file slaqr0.f.

  subroutine slaqr1 (integer N, real, dimension( ldh, * ) H, integer LDH, real SR1, real SI1, real SR2, real SI2, real, dimension( * ) V)

SLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.

Purpose:

          Given a 2-by-2 or 3-by-3 matrix H, SLAQR1 sets v to a
          scalar multiple of the first column of the product
          (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
          scaling to avoid overflows and most underflows. It
          is assumed that either
                  1) sr1 = sr2 and si1 = -si2
              or
                  2) si1 = si2 = 0.
          This is useful for starting double implicit shift bulges
          in the QR algorithm.

Parameters
    N

              N is INTEGER
                  Order of the matrix H. N must be either 2 or 3.

    H

              H is REAL array, dimension (LDH,N)
                  The 2-by-2 or 3-by-3 matrix H in (*).

    LDH

              LDH is INTEGER
                  The leading dimension of H as declared in
                  the calling procedure.  LDH >= N

    SR1

              SR1 is REAL

    SI1

              SI1 is REAL

    SR2

              SR2 is REAL

    SI2

              SI2 is REAL
                  The shifts in (*).

    V

              V is REAL array, dimension (N)
                  A scalar multiple of the first column of the
                  matrix K in (*).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2017

Contributors:
    Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 123 of file slaqr1.f.

  subroutine slaqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, real, dimension( * ) SR, real, dimension( * ) SI, real, dimension( ldv, * ) V, integer LDV, integer NH, real, dimension( ldt, * ) T, integer LDT, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, real, dimension( * ) WORK, integer LWORK)

SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Purpose:

        SLAQR2 is identical to SLAQR3 except that it avoids
        recursion by calling SLAHQR instead of SLAQR4.
        Aggressive early deflation:
        This subroutine accepts as input an upper Hessenberg matrix
        H and performs an orthogonal similarity transformation
        designed to detect and deflate fully converged eigenvalues from
        a trailing principal submatrix.  On output H has been over-
        written by a new Hessenberg matrix that is a perturbation of
        an orthogonal similarity transformation of H.  It is to be
        hoped that the final version of H has many zero subdiagonal
        entries.

Parameters
    WANTT

              WANTT is LOGICAL
              If .TRUE., then the Hessenberg matrix H is fully updated
              so that the quasi-triangular Schur factor may be
              computed (in cooperation with the calling subroutine).
              If .FALSE., then only enough of H is updated to preserve
              the eigenvalues.

    WANTZ

              WANTZ is LOGICAL
              If .TRUE., then the orthogonal matrix Z is updated so
              so that the orthogonal Schur factor may be computed
              (in cooperation with the calling subroutine).
              If .FALSE., then Z is not referenced.

    N

              N is INTEGER
              The order of the matrix H and (if WANTZ is .TRUE.) the
              order of the orthogonal matrix Z.

    KTOP

              KTOP is INTEGER
              It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
              KBOT and KTOP together determine an isolated block
              along the diagonal of the Hessenberg matrix.

    KBOT

              KBOT is INTEGER
              It is assumed without a check that either
              KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
              determine an isolated block along the diagonal of the
              Hessenberg matrix.

    NW

              NW is INTEGER
              Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

    H

              H is REAL array, dimension (LDH,N)
              On input the initial N-by-N section of H stores the
              Hessenberg matrix undergoing aggressive early deflation.
              On output H has been transformed by an orthogonal
              similarity transformation, perturbed, and the returned
              to Hessenberg form that (it is to be hoped) has some
              zero subdiagonal entries.

    LDH

              LDH is INTEGER
              Leading dimension of H just as declared in the calling
              subroutine.  N <= LDH

    ILOZ

              ILOZ is INTEGER

    IHIZ

              IHIZ is INTEGER
              Specify the rows of Z to which transformations must be
              applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

    Z

              Z is REAL array, dimension (LDZ,N)
              IF WANTZ is .TRUE., then on output, the orthogonal
              similarity transformation mentioned above has been
              accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
              If WANTZ is .FALSE., then Z is unreferenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of Z just as declared in the
              calling subroutine.  1 <= LDZ.

    NS

              NS is INTEGER
              The number of unconverged (ie approximate) eigenvalues
              returned in SR and SI that may be used as shifts by the
              calling subroutine.

    ND

              ND is INTEGER
              The number of converged eigenvalues uncovered by this
              subroutine.

    SR

              SR is REAL array, dimension (KBOT)

    SI

              SI is REAL array, dimension (KBOT)
              On output, the real and imaginary parts of approximate
              eigenvalues that may be used for shifts are stored in
              SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
              SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
              The real and imaginary parts of converged eigenvalues
              are stored in SR(KBOT-ND+1) through SR(KBOT) and
              SI(KBOT-ND+1) through SI(KBOT), respectively.

    V

              V is REAL array, dimension (LDV,NW)
              An NW-by-NW work array.

    LDV

              LDV is INTEGER
              The leading dimension of V just as declared in the
              calling subroutine.  NW <= LDV

    NH

              NH is INTEGER
              The number of columns of T.  NH >= NW.

    T

              T is REAL array, dimension (LDT,NW)

    LDT

              LDT is INTEGER
              The leading dimension of T just as declared in the
              calling subroutine.  NW <= LDT

    NV

              NV is INTEGER
              The number of rows of work array WV available for
              workspace.  NV >= NW.

    WV

              WV is REAL array, dimension (LDWV,NW)

    LDWV

              LDWV is INTEGER
              The leading dimension of W just as declared in the
              calling subroutine.  NW <= LDV

    WORK

              WORK is REAL array, dimension (LWORK)
              On exit, WORK(1) is set to an estimate of the optimal value
              of LWORK for the given values of N, NW, KTOP and KBOT.

    LWORK

              LWORK is INTEGER
              The dimension of the work array WORK.  LWORK = 2*NW
              suffices, but greater efficiency may result from larger
              values of LWORK.
              If LWORK = -1, then a workspace query is assumed; SLAQR2
              only estimates the optimal workspace size for the given
              values of N, NW, KTOP and KBOT.  The estimate is returned
              in WORK(1).  No error message related to LWORK is issued
              by XERBLA.  Neither H nor Z are accessed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2017

Contributors:
    Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 280 of file slaqr2.f.

  subroutine slaqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, real, dimension( * ) SR, real, dimension( * ) SI, real, dimension( ldv, * ) V, integer LDV, integer NH, real, dimension( ldt, * ) T, integer LDT, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, real, dimension( * ) WORK, integer LWORK)

SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Purpose:

        Aggressive early deflation:
        SLAQR3 accepts as input an upper Hessenberg matrix
        H and performs an orthogonal similarity transformation
        designed to detect and deflate fully converged eigenvalues from
        a trailing principal submatrix.  On output H has been over-
        written by a new Hessenberg matrix that is a perturbation of
        an orthogonal similarity transformation of H.  It is to be
        hoped that the final version of H has many zero subdiagonal
        entries.

Parameters
    WANTT

              WANTT is LOGICAL
              If .TRUE., then the Hessenberg matrix H is fully updated
              so that the quasi-triangular Schur factor may be
              computed (in cooperation with the calling subroutine).
              If .FALSE., then only enough of H is updated to preserve
              the eigenvalues.

    WANTZ

              WANTZ is LOGICAL
              If .TRUE., then the orthogonal matrix Z is updated so
              so that the orthogonal Schur factor may be computed
              (in cooperation with the calling subroutine).
              If .FALSE., then Z is not referenced.

    N

              N is INTEGER
              The order of the matrix H and (if WANTZ is .TRUE.) the
              order of the orthogonal matrix Z.

    KTOP

              KTOP is INTEGER
              It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
              KBOT and KTOP together determine an isolated block
              along the diagonal of the Hessenberg matrix.

    KBOT

              KBOT is INTEGER
              It is assumed without a check that either
              KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
              determine an isolated block along the diagonal of the
              Hessenberg matrix.

    NW

              NW is INTEGER
              Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

    H

              H is REAL array, dimension (LDH,N)
              On input the initial N-by-N section of H stores the
              Hessenberg matrix undergoing aggressive early deflation.
              On output H has been transformed by an orthogonal
              similarity transformation, perturbed, and the returned
              to Hessenberg form that (it is to be hoped) has some
              zero subdiagonal entries.

    LDH

              LDH is INTEGER
              Leading dimension of H just as declared in the calling
              subroutine.  N <= LDH

    ILOZ

              ILOZ is INTEGER

    IHIZ

              IHIZ is INTEGER
              Specify the rows of Z to which transformations must be
              applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

    Z

              Z is REAL array, dimension (LDZ,N)
              IF WANTZ is .TRUE., then on output, the orthogonal
              similarity transformation mentioned above has been
              accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
              If WANTZ is .FALSE., then Z is unreferenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of Z just as declared in the
              calling subroutine.  1 <= LDZ.

    NS

              NS is INTEGER
              The number of unconverged (ie approximate) eigenvalues
              returned in SR and SI that may be used as shifts by the
              calling subroutine.

    ND

              ND is INTEGER
              The number of converged eigenvalues uncovered by this
              subroutine.

    SR

              SR is REAL array, dimension (KBOT)

    SI

              SI is REAL array, dimension (KBOT)
              On output, the real and imaginary parts of approximate
              eigenvalues that may be used for shifts are stored in
              SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
              SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
              The real and imaginary parts of converged eigenvalues
              are stored in SR(KBOT-ND+1) through SR(KBOT) and
              SI(KBOT-ND+1) through SI(KBOT), respectively.

    V

              V is REAL array, dimension (LDV,NW)
              An NW-by-NW work array.

    LDV

              LDV is INTEGER
              The leading dimension of V just as declared in the
              calling subroutine.  NW <= LDV

    NH

              NH is INTEGER
              The number of columns of T.  NH >= NW.

    T

              T is REAL array, dimension (LDT,NW)

    LDT

              LDT is INTEGER
              The leading dimension of T just as declared in the
              calling subroutine.  NW <= LDT

    NV

              NV is INTEGER
              The number of rows of work array WV available for
              workspace.  NV >= NW.

    WV

              WV is REAL array, dimension (LDWV,NW)

    LDWV

              LDWV is INTEGER
              The leading dimension of W just as declared in the
              calling subroutine.  NW <= LDV

    WORK

              WORK is REAL array, dimension (LWORK)
              On exit, WORK(1) is set to an estimate of the optimal value
              of LWORK for the given values of N, NW, KTOP and KBOT.

    LWORK

              LWORK is INTEGER
              The dimension of the work array WORK.  LWORK = 2*NW
              suffices, but greater efficiency may result from larger
              values of LWORK.
              If LWORK = -1, then a workspace query is assumed; SLAQR3
              only estimates the optimal workspace size for the given
              values of N, NW, KTOP and KBOT.  The estimate is returned
              in WORK(1).  No error message related to LWORK is issued
              by XERBLA.  Neither H nor Z are accessed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2016

Contributors:
    Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 277 of file slaqr3.f.

  subroutine slaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)

SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

        SLAQR4 implements one level of recursion for SLAQR0.
        It is a complete implementation of the small bulge multi-shift
        QR algorithm.  It may be called by SLAQR0 and, for large enough
        deflation window size, it may be called by SLAQR3.  This
        subroutine is identical to SLAQR0 except that it calls SLAQR2
        instead of SLAQR3.
        SLAQR4 computes the eigenvalues of a Hessenberg matrix H
        and, optionally, the matrices T and Z from the Schur decomposition
        H = Z T Z**T, where T is an upper quasi-triangular matrix (the
        Schur form), and Z is the orthogonal matrix of Schur vectors.
        Optionally Z may be postmultiplied into an input orthogonal
        matrix Q so that this routine can give the Schur factorization
        of a matrix A which has been reduced to the Hessenberg form H
        by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

Parameters
    WANTT

              WANTT is LOGICAL
              = .TRUE. : the full Schur form T is required;
              = .FALSE.: only eigenvalues are required.

    WANTZ

              WANTZ is LOGICAL
              = .TRUE. : the matrix of Schur vectors Z is required;
              = .FALSE.: Schur vectors are not required.

    N

              N is INTEGER
               The order of the matrix H.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
               It is assumed that H is already upper triangular in rows
               and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
               H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
               previous call to SGEBAL, and then passed to SGEHRD when the
               matrix output by SGEBAL is reduced to Hessenberg form.
               Otherwise, ILO and IHI should be set to 1 and N,
               respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
               If N = 0, then ILO = 1 and IHI = 0.

    H

              H is REAL array, dimension (LDH,N)
               On entry, the upper Hessenberg matrix H.
               On exit, if INFO = 0 and WANTT is .TRUE., then H contains
               the upper quasi-triangular matrix T from the Schur
               decomposition (the Schur form); 2-by-2 diagonal blocks
               (corresponding to complex conjugate pairs of eigenvalues)
               are returned in standard form, with H(i,i) = H(i+1,i+1)
               and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
               .FALSE., then the contents of H are unspecified on exit.
               (The output value of H when INFO > 0 is given under the
               description of INFO below.)
               This subroutine may explicitly set H(i,j) = 0 for i > j and
               j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

    LDH

              LDH is INTEGER
               The leading dimension of the array H. LDH >= max(1,N).

    WR

              WR is REAL array, dimension (IHI)

    WI

              WI is REAL array, dimension (IHI)
               The real and imaginary parts, respectively, of the computed
               eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
               and WI(ILO:IHI). If two eigenvalues are computed as a
               complex conjugate pair, they are stored in consecutive
               elements of WR and WI, say the i-th and (i+1)th, with
               WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
               the eigenvalues are stored in the same order as on the
               diagonal of the Schur form returned in H, with
               WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
               block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
               WI(i+1) = -WI(i).

    ILOZ

              ILOZ is INTEGER

    IHIZ

              IHIZ is INTEGER
               Specify the rows of Z to which transformations must be
               applied if WANTZ is .TRUE..
               1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

    Z

              Z is REAL array, dimension (LDZ,IHI)
               If WANTZ is .FALSE., then Z is not referenced.
               If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
               replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
               orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
               (The output value of Z when INFO > 0 is given under
               the description of INFO below.)

    LDZ

              LDZ is INTEGER
               The leading dimension of the array Z.  if WANTZ is .TRUE.
               then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

    WORK

              WORK is REAL array, dimension LWORK
               On exit, if LWORK = -1, WORK(1) returns an estimate of
               the optimal value for LWORK.

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK.  LWORK >= max(1,N)
               is sufficient, but LWORK typically as large as 6*N may
               be required for optimal performance.  A workspace query
               to determine the optimal workspace size is recommended.
               If LWORK = -1, then SLAQR4 does a workspace query.
               In this case, SLAQR4 checks the input parameters and
               estimates the optimal workspace size for the given
               values of N, ILO and IHI.  The estimate is returned
               in WORK(1).  No error message related to LWORK is
               issued by XERBLA.  Neither H nor Z are accessed.

    INFO

              INFO is INTEGER

              INFO is INTEGER
                 = 0:  successful exit
                 > 0:  if INFO = i, SLAQR4 failed to compute all of
                    the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                    and WI contain those eigenvalues which have been
                    successfully computed.  (Failures are rare.)
                    If INFO > 0 and WANT is .FALSE., then on exit,
                    the remaining unconverged eigenvalues are the eigen-
                    values of the upper Hessenberg matrix rows and
                    columns ILO through INFO of the final, output
                    value of H.
                    If INFO > 0 and WANTT is .TRUE., then on exit
               (*)  (initial value of H)*U  = U*(final value of H)
                    where U is a orthogonal matrix.  The final
                    value of  H is upper Hessenberg and triangular in
                    rows and columns INFO+1 through IHI.
                    If INFO > 0 and WANTZ is .TRUE., then on exit
                      (final value of Z(ILO:IHI,ILOZ:IHIZ)
                       =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
                    where U is the orthogonal matrix in (*) (regard-
                    less of the value of WANTT.)
                    If INFO > 0 and WANTZ is .FALSE., then Z is not
                    accessed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Contributors:
    Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

      K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
      Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
      Performance, SIAM Journal of Matrix Analysis, volume 23, pages
      929--947, 2002.

     K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 267 of file slaqr4.f.

  subroutine slaqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP, integer KBOT, integer NSHFTS, real, dimension( * ) SR, real, dimension( * ) SI, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldu, * ) U, integer LDU, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, integer NH, real, dimension( ldwh, * ) WH, integer LDWH)

SLAQR5 performs a single small-bulge multi-shift QR sweep.

Purpose:

        SLAQR5, called by SLAQR0, performs a
        single small-bulge multi-shift QR sweep.

Parameters
    WANTT

              WANTT is LOGICAL
                 WANTT = .true. if the quasi-triangular Schur factor
                 is being computed.  WANTT is set to .false. otherwise.

    WANTZ

              WANTZ is LOGICAL
                 WANTZ = .true. if the orthogonal Schur factor is being
                 computed.  WANTZ is set to .false. otherwise.

    KACC22

              KACC22 is INTEGER with value 0, 1, or 2.
                 Specifies the computation mode of far-from-diagonal
                 orthogonal updates.
            = 0: SLAQR5 does not accumulate reflections and does not
                 use matrix-matrix multiply to update far-from-diagonal
                 matrix entries.
            = 1: SLAQR5 accumulates reflections and uses matrix-matrix
                 multiply to update the far-from-diagonal matrix entries.
            = 2: SLAQR5 accumulates reflections, uses matrix-matrix
                 multiply to update the far-from-diagonal matrix entries,
                 and takes advantage of 2-by-2 block structure during
                 matrix multiplies.

    N

              N is INTEGER
                 N is the order of the Hessenberg matrix H upon which this
                 subroutine operates.

    KTOP

              KTOP is INTEGER

    KBOT

              KBOT is INTEGER
                 These are the first and last rows and columns of an
                 isolated diagonal block upon which the QR sweep is to be
                 applied. It is assumed without a check that
                           either KTOP = 1  or   H(KTOP,KTOP-1) = 0
                 and
                           either KBOT = N  or   H(KBOT+1,KBOT) = 0.

    NSHFTS

              NSHFTS is INTEGER
                 NSHFTS gives the number of simultaneous shifts.  NSHFTS
                 must be positive and even.

    SR

              SR is REAL array, dimension (NSHFTS)

    SI

              SI is REAL array, dimension (NSHFTS)
                 SR contains the real parts and SI contains the imaginary
                 parts of the NSHFTS shifts of origin that define the
                 multi-shift QR sweep.  On output SR and SI may be
                 reordered.

    H

              H is REAL array, dimension (LDH,N)
                 On input H contains a Hessenberg matrix.  On output a
                 multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
                 to the isolated diagonal block in rows and columns KTOP
                 through KBOT.

    LDH

              LDH is INTEGER
                 LDH is the leading dimension of H just as declared in the
                 calling procedure.  LDH >= MAX(1,N).

    ILOZ

              ILOZ is INTEGER

    IHIZ

              IHIZ is INTEGER
                 Specify the rows of Z to which transformations must be
                 applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N

    Z

              Z is REAL array, dimension (LDZ,IHIZ)
                 If WANTZ = .TRUE., then the QR Sweep orthogonal
                 similarity transformation is accumulated into
                 Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
                 If WANTZ = .FALSE., then Z is unreferenced.

    LDZ

              LDZ is INTEGER
                 LDA is the leading dimension of Z just as declared in
                 the calling procedure. LDZ >= N.

    V

              V is REAL array, dimension (LDV,NSHFTS/2)

    LDV

              LDV is INTEGER
                 LDV is the leading dimension of V as declared in the
                 calling procedure.  LDV >= 3.

    U

              U is REAL array, dimension (LDU,3*NSHFTS-3)

    LDU

              LDU is INTEGER
                 LDU is the leading dimension of U just as declared in the
                 in the calling subroutine.  LDU >= 3*NSHFTS-3.

    NV

              NV is INTEGER
                 NV is the number of rows in WV agailable for workspace.
                 NV >= 1.

    WV

              WV is REAL array, dimension (LDWV,3*NSHFTS-3)

    LDWV

              LDWV is INTEGER
                 LDWV is the leading dimension of WV as declared in the
                 in the calling subroutine.  LDWV >= NV.

    NH

              NH is INTEGER
                 NH is the number of columns in array WH available for
                 workspace. NH >= 1.

    WH

              WH is REAL array, dimension (LDWH,NH)

    LDWH

              LDWH is INTEGER
                 Leading dimension of WH just as declared in the
                 calling procedure.  LDWH >= 3*NSHFTS-3.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2016

Contributors:
    Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:
    K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

Definition at line 259 of file slaqr5.f.

  subroutine slaqsb (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

SLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.

Purpose:

     SLAQSB equilibrates a symmetric band matrix A using the scaling
     factors in the vector S.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of super-diagonals of the matrix A if UPLO = 'U',
              or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, if INFO = 0, the triangular factor U or L from the
              Cholesky factorization A = U**T*U or A = L*L**T of the band
              matrix A, in the same storage format as A.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    S

              S is REAL array, dimension (N)
              The scale factors for A.

    SCOND

              SCOND is REAL
              Ratio of the smallest S(i) to the largest S(i).

    AMAX

              AMAX is REAL
              Absolute value of largest matrix entry.

    EQUED

              EQUED is CHARACTER*1
              Specifies whether or not equilibration was done.
              = 'N':  No equilibration.
              = 'Y':  Equilibration was done, i.e., A has been replaced by
                      diag(S) * A * diag(S).

Internal Parameters:

      THRESH is a threshold value used to decide if scaling should be done
      based on the ratio of the scaling factors.  If SCOND < THRESH,
      scaling is done.
      LARGE and SMALL are threshold values used to decide if scaling should
      be done based on the absolute size of the largest matrix element.
      If AMAX > LARGE or AMAX < SMALL, scaling is done.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 142 of file slaqsb.f.

  subroutine slaqsp (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

SLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.

Purpose:

     SLAQSP equilibrates a symmetric matrix A using the scaling factors
     in the vector S.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
              the same storage format as A.

    S

              S is REAL array, dimension (N)
              The scale factors for A.

    SCOND

              SCOND is REAL
              Ratio of the smallest S(i) to the largest S(i).

    AMAX

              AMAX is REAL
              Absolute value of largest matrix entry.

    EQUED

              EQUED is CHARACTER*1
              Specifies whether or not equilibration was done.
              = 'N':  No equilibration.
              = 'Y':  Equilibration was done, i.e., A has been replaced by
                      diag(S) * A * diag(S).

Internal Parameters:

      THRESH is a threshold value used to decide if scaling should be done
      based on the ratio of the scaling factors.  If SCOND < THRESH,
      scaling is done.
      LARGE and SMALL are threshold values used to decide if scaling should
      be done based on the absolute size of the largest matrix element.
      If AMAX > LARGE or AMAX < SMALL, scaling is done.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 127 of file slaqsp.f.

  subroutine slaqtr (logical LTRAN, logical LREAL, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) B, real W, real SCALE, real, dimension( * ) X, real, dimension( * ) WORK, integer INFO)

SLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.

Purpose:

     SLAQTR solves the real quasi-triangular system
                  op(T)*p = scale*c,               if LREAL = .TRUE.
     or the complex quasi-triangular systems
                op(T + iB)*(p+iq) = scale*(c+id),  if LREAL = .FALSE.
     in real arithmetic, where T is upper quasi-triangular.
     If LREAL = .FALSE., then the first diagonal block of T must be
     1 by 1, B is the specially structured matrix
                    B = [ b(1) b(2) ... b(n) ]
                        [       w            ]
                        [           w        ]
                        [              .     ]
                        [                 w  ]
     op(A) = A or A**T, A**T denotes the transpose of
     matrix A.
     On input, X = [ c ].  On output, X = [ p ].
                   [ d ]                  [ q ]
     This subroutine is designed for the condition number estimation
     in routine STRSNA.

Parameters
    LTRAN

              LTRAN is LOGICAL
              On entry, LTRAN specifies the option of conjugate transpose:
                 = .FALSE.,    op(T+i*B) = T+i*B,
                 = .TRUE.,     op(T+i*B) = (T+i*B)**T.

    LREAL

              LREAL is LOGICAL
              On entry, LREAL specifies the input matrix structure:
                 = .FALSE.,    the input is complex
                 = .TRUE.,     the input is real

    N

              N is INTEGER
              On entry, N specifies the order of T+i*B. N >= 0.

    T

              T is REAL array, dimension (LDT,N)
              On entry, T contains a matrix in Schur canonical form.
              If LREAL = .FALSE., then the first diagonal block of T must
              be 1 by 1.

    LDT

              LDT is INTEGER
              The leading dimension of the matrix T. LDT >= max(1,N).

    B

              B is REAL array, dimension (N)
              On entry, B contains the elements to form the matrix
              B as described above.
              If LREAL = .TRUE., B is not referenced.

    W

              W is REAL
              On entry, W is the diagonal element of the matrix B.
              If LREAL = .TRUE., W is not referenced.

    SCALE

              SCALE is REAL
              On exit, SCALE is the scale factor.

    X

              X is REAL array, dimension (2*N)
              On entry, X contains the right hand side of the system.
              On exit, X is overwritten by the solution.

    WORK

              WORK is REAL array, dimension (N)

    INFO

              INFO is INTEGER
              On exit, INFO is set to
                 0: successful exit.
                   1: the some diagonal 1 by 1 block has been perturbed by
                      a small number SMIN to keep nonsingularity.
                   2: the some diagonal 2 by 2 block has been perturbed by
                      a small number in SLALN2 to keep nonsingularity.
              NOTE: In the interests of speed, this routine does not
                    check the inputs for errors.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 167 of file slaqtr.f.

  subroutine slar1v (integer N, integer B1, integer BN, real LAMBDA, real, dimension( * ) D, real, dimension( * ) L, real, dimension( * ) LD, real, dimension( * ) LLD, real PIVMIN, real GAPTOL, real, dimension( * ) Z, logical WANTNC, integer NEGCNT, real ZTZ, real MINGMA, integer R, integer, dimension( * ) ISUPPZ, real NRMINV, real RESID, real RQCORR, real, dimension( * ) WORK)

SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

Purpose:

     SLAR1V computes the (scaled) r-th column of the inverse of
     the sumbmatrix in rows B1 through BN of the tridiagonal matrix
     L D L**T - sigma I. When sigma is close to an eigenvalue, the
     computed vector is an accurate eigenvector. Usually, r corresponds
     to the index where the eigenvector is largest in magnitude.
     The following steps accomplish this computation :
     (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
     (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
     (c) Computation of the diagonal elements of the inverse of
         L D L**T - sigma I by combining the above transforms, and choosing
         r as the index where the diagonal of the inverse is (one of the)
         largest in magnitude.
     (d) Computation of the (scaled) r-th column of the inverse using the
         twisted factorization obtained by combining the top part of the
         the stationary and the bottom part of the progressive transform.

Parameters
    N

              N is INTEGER
               The order of the matrix L D L**T.

    B1

              B1 is INTEGER
               First index of the submatrix of L D L**T.

    BN

              BN is INTEGER
               Last index of the submatrix of L D L**T.

    LAMBDA

              LAMBDA is REAL
               The shift. In order to compute an accurate eigenvector,
               LAMBDA should be a good approximation to an eigenvalue
               of L D L**T.

    L

              L is REAL array, dimension (N-1)
               The (n-1) subdiagonal elements of the unit bidiagonal matrix
               L, in elements 1 to N-1.

    D

              D is REAL array, dimension (N)
               The n diagonal elements of the diagonal matrix D.

    LD

              LD is REAL array, dimension (N-1)
               The n-1 elements L(i)*D(i).

    LLD

              LLD is REAL array, dimension (N-1)
               The n-1 elements L(i)*L(i)*D(i).

    PIVMIN

              PIVMIN is REAL
               The minimum pivot in the Sturm sequence.

    GAPTOL

              GAPTOL is REAL
               Tolerance that indicates when eigenvector entries are negligible
               w.r.t. their contribution to the residual.

    Z

              Z is REAL array, dimension (N)
               On input, all entries of Z must be set to 0.
               On output, Z contains the (scaled) r-th column of the
               inverse. The scaling is such that Z(R) equals 1.

    WANTNC

              WANTNC is LOGICAL
               Specifies whether NEGCNT has to be computed.

    NEGCNT

              NEGCNT is INTEGER
               If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
               in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.

    ZTZ

              ZTZ is REAL
               The square of the 2-norm of Z.

    MINGMA

              MINGMA is REAL
               The reciprocal of the largest (in magnitude) diagonal
               element of the inverse of L D L**T - sigma I.

    R

              R is INTEGER
               The twist index for the twisted factorization used to
               compute Z.
               On input, 0 <= R <= N. If R is input as 0, R is set to
               the index where (L D L**T - sigma I)^{-1} is largest
               in magnitude. If 1 <= R <= N, R is unchanged.
               On output, R contains the twist index used to compute Z.
               Ideally, R designates the position of the maximum entry in the
               eigenvector.

    ISUPPZ

              ISUPPZ is INTEGER array, dimension (2)
               The support of the vector in Z, i.e., the vector Z is
               nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

    NRMINV

              NRMINV is REAL
               NRMINV = 1/SQRT( ZTZ )

    RESID

              RESID is REAL
               The residual of the FP vector.
               RESID = ABS( MINGMA )/SQRT( ZTZ )

    RQCORR

              RQCORR is REAL
               The Rayleigh Quotient correction to LAMBDA.
               RQCORR = MINGMA*TMP

    WORK

              WORK is REAL array, dimension (4*N)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

Definition at line 232 of file slar1v.f.

  subroutine slar2v (integer N, real, dimension( * ) X, real, dimension( * ) Y, real, dimension( * ) Z, integer INCX, real, dimension( * ) C, real, dimension( * ) S, integer INCC)

SLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.

Purpose:

     SLAR2V applies a vector of real plane rotations from both sides to
     a sequence of 2-by-2 real symmetric matrices, defined by the elements
     of the vectors x, y and z. For i = 1,2,...,n
        ( x(i)  z(i) ) := (  c(i)  s(i) ) ( x(i)  z(i) ) ( c(i) -s(i) )
        ( z(i)  y(i) )    ( -s(i)  c(i) ) ( z(i)  y(i) ) ( s(i)  c(i) )

Parameters
    N

              N is INTEGER
              The number of plane rotations to be applied.

    X

              X is REAL array,
                             dimension (1+(N-1)*INCX)
              The vector x.

    Y

              Y is REAL array,
                             dimension (1+(N-1)*INCX)
              The vector y.

    Z

              Z is REAL array,
                             dimension (1+(N-1)*INCX)
              The vector z.

    INCX

              INCX is INTEGER
              The increment between elements of X, Y and Z. INCX > 0.

    C

              C is REAL array, dimension (1+(N-1)*INCC)
              The cosines of the plane rotations.

    S

              S is REAL array, dimension (1+(N-1)*INCC)
              The sines of the plane rotations.

    INCC

              INCC is INTEGER
              The increment between elements of C and S. INCC > 0.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 112 of file slar2v.f.

  subroutine slarf (character SIDE, integer M, integer N, real, dimension( * ) V, integer INCV, real TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK)

SLARF applies an elementary reflector to a general rectangular matrix.

Purpose:

     SLARF applies a real elementary reflector H to a real m by n matrix
     C, from either the left or the right. H is represented in the form
           H = I - tau * v * v**T
     where tau is a real scalar and v is a real vector.
     If tau = 0, then H is taken to be the unit matrix.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': form  H * C
              = 'R': form  C * H

    M

              M is INTEGER
              The number of rows of the matrix C.

    N

              N is INTEGER
              The number of columns of the matrix C.

    V

              V is REAL array, dimension
                         (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                      or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
              The vector v in the representation of H. V is not used if
              TAU = 0.

    INCV

              INCV is INTEGER
              The increment between elements of v. INCV <> 0.

    TAU

              TAU is REAL
              The value tau in the representation of H.

    C

              C is REAL array, dimension (LDC,N)
              On entry, the m by n matrix C.
              On exit, C is overwritten by the matrix H * C if SIDE = 'L',
              or C * H if SIDE = 'R'.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is REAL array, dimension
                             (N) if SIDE = 'L'
                          or (M) if SIDE = 'R'

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 126 of file slarf.f.

  subroutine slarfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integer LDC, real, dimension( ldwork, * ) WORK, integer LDWORK)

SLARFB applies a block reflector or its transpose to a general rectangular matrix.

Purpose:

     SLARFB applies a real block reflector H or its transpose H**T to a
     real m by n matrix C, from either the left or the right.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply H or H**T from the Left
              = 'R': apply H or H**T from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply H (No transpose)
              = 'T': apply H**T (Transpose)

    DIRECT

              DIRECT is CHARACTER*1
              Indicates how H is formed from a product of elementary
              reflectors
              = 'F': H = H(1) H(2) . . . H(k) (Forward)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Indicates how the vectors which define the elementary
              reflectors are stored:
              = 'C': Columnwise
              = 'R': Rowwise

    M

              M is INTEGER
              The number of rows of the matrix C.

    N

              N is INTEGER
              The number of columns of the matrix C.

    K

              K is INTEGER
              The order of the matrix T (= the number of elementary
              reflectors whose product defines the block reflector).
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    V

              V is REAL array, dimension
                                    (LDV,K) if STOREV = 'C'
                                    (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                    (LDV,N) if STOREV = 'R' and SIDE = 'R'
              The matrix V. See Further Details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
              if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
              if STOREV = 'R', LDV >= K.

    T

              T is REAL array, dimension (LDT,K)
              The triangular k by k matrix T in the representation of the
              block reflector.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

    C

              C is REAL array, dimension (LDC,N)
              On entry, the m by n matrix C.
              On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is REAL array, dimension (LDWORK,K)

    LDWORK

              LDWORK is INTEGER
              The leading dimension of the array WORK.
              If SIDE = 'L', LDWORK >= max(1,N);
              if SIDE = 'R', LDWORK >= max(1,M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2013

Further Details:

      The shape of the matrix V and the storage of the vectors which define
      the H(i) is best illustrated by the following example with n = 5 and
      k = 3. The elements equal to 1 are not stored; the corresponding
      array elements are modified but restored on exit. The rest of the
      array is not used.
      DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                   V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                       ( v1  1    )                     (     1 v2 v2 v2 )
                       ( v1 v2  1 )                     (        1 v3 v3 )
                       ( v1 v2 v3 )
                       ( v1 v2 v3 )
      DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                   V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                       ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                       (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                       (     1 v3 )
                       (        1 )

Definition at line 199 of file slarfb.f.

  subroutine slarfg (integer N, real ALPHA, real, dimension( * ) X, integer INCX, real TAU)

SLARFG generates an elementary reflector (Householder matrix).

Purpose:

     SLARFG generates a real elementary reflector H of order n, such
     that
           H * ( alpha ) = ( beta ),   H**T * H = I.
               (   x   )   (   0  )
     where alpha and beta are scalars, and x is an (n-1)-element real
     vector. H is represented in the form
           H = I - tau * ( 1 ) * ( 1 v**T ) ,
                         ( v )
     where tau is a real scalar and v is a real (n-1)-element
     vector.
     If the elements of x are all zero, then tau = 0 and H is taken to be
     the unit matrix.
     Otherwise  1 <= tau <= 2.

Parameters
    N

              N is INTEGER
              The order of the elementary reflector.

    ALPHA

              ALPHA is REAL
              On entry, the value alpha.
              On exit, it is overwritten with the value beta.

    X

              X is REAL array, dimension
                             (1+(N-2)*abs(INCX))
              On entry, the vector x.
              On exit, it is overwritten with the vector v.

    INCX

              INCX is INTEGER
              The increment between elements of X. INCX > 0.

    TAU

              TAU is REAL
              The value tau.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    November 2017

Definition at line 108 of file slarfg.f.

  subroutine slarfgp (integer N, real ALPHA, real, dimension( * ) X, integer INCX, real TAU)

SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

Purpose:

     SLARFGP generates a real elementary reflector H of order n, such
     that
           H * ( alpha ) = ( beta ),   H**T * H = I.
               (   x   )   (   0  )
     where alpha and beta are scalars, beta is non-negative, and x is
     an (n-1)-element real vector.  H is represented in the form
           H = I - tau * ( 1 ) * ( 1 v**T ) ,
                         ( v )
     where tau is a real scalar and v is a real (n-1)-element
     vector.
     If the elements of x are all zero, then tau = 0 and H is taken to be
     the unit matrix.

Parameters
    N

              N is INTEGER
              The order of the elementary reflector.

    ALPHA

              ALPHA is REAL
              On entry, the value alpha.
              On exit, it is overwritten with the value beta.

    X

              X is REAL array, dimension
                             (1+(N-2)*abs(INCX))
              On entry, the vector x.
              On exit, it is overwritten with the vector v.

    INCX

              INCX is INTEGER
              The increment between elements of X. INCX > 0.

    TAU

              TAU is REAL
              The value tau.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    November 2017

Definition at line 106 of file slarfgp.f.

  subroutine slarft (character DIRECT, character STOREV, integer N, integer K, real, dimension( ldv, * ) V, integer LDV, real, dimension( * ) TAU, real, dimension( ldt, * ) T, integer LDT)

SLARFT forms the triangular factor T of a block reflector H = I - vtvH

Purpose:

     SLARFT forms the triangular factor T of a real block reflector H
     of order n, which is defined as a product of k elementary reflectors.
     If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
     If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
     If STOREV = 'C', the vector which defines the elementary reflector
     H(i) is stored in the i-th column of the array V, and
        H  =  I - V * T * V**T
     If STOREV = 'R', the vector which defines the elementary reflector
     H(i) is stored in the i-th row of the array V, and
        H  =  I - V**T * T * V

Parameters
    DIRECT

              DIRECT is CHARACTER*1
              Specifies the order in which the elementary reflectors are
              multiplied to form the block reflector:
              = 'F': H = H(1) H(2) . . . H(k) (Forward)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Specifies how the vectors which define the elementary
              reflectors are stored (see also Further Details):
              = 'C': columnwise
              = 'R': rowwise

    N

              N is INTEGER
              The order of the block reflector H. N >= 0.

    K

              K is INTEGER
              The order of the triangular factor T (= the number of
              elementary reflectors). K >= 1.

    V

              V is REAL array, dimension
                                   (LDV,K) if STOREV = 'C'
                                   (LDV,N) if STOREV = 'R'
              The matrix V. See further details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

    TAU

              TAU is REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i).

    T

              T is REAL array, dimension (LDT,K)
              The k by k triangular factor T of the block reflector.
              If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
              lower triangular. The rest of the array is not used.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      The shape of the matrix V and the storage of the vectors which define
      the H(i) is best illustrated by the following example with n = 5 and
      k = 3. The elements equal to 1 are not stored.
      DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                   V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                       ( v1  1    )                     (     1 v2 v2 v2 )
                       ( v1 v2  1 )                     (        1 v3 v3 )
                       ( v1 v2 v3 )
                       ( v1 v2 v3 )
      DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                   V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                       ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                       (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                       (     1 v3 )
                       (        1 )

Definition at line 165 of file slarft.f.

  subroutine slarfx (character SIDE, integer M, integer N, real, dimension( * ) V, real TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK)

SLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.

Purpose:

     SLARFX applies a real elementary reflector H to a real m by n
     matrix C, from either the left or the right. H is represented in the
     form
           H = I - tau * v * v**T
     where tau is a real scalar and v is a real vector.
     If tau = 0, then H is taken to be the unit matrix
     This version uses inline code if H has order < 11.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': form  H * C
              = 'R': form  C * H

    M

              M is INTEGER
              The number of rows of the matrix C.

    N

              N is INTEGER
              The number of columns of the matrix C.

    V

              V is REAL array, dimension (M) if SIDE = 'L'
                                         or (N) if SIDE = 'R'
              The vector v in the representation of H.

    TAU

              TAU is REAL
              The value tau in the representation of H.

    C

              C is REAL array, dimension (LDC,N)
              On entry, the m by n matrix C.
              On exit, C is overwritten by the matrix H * C if SIDE = 'L',
              or C * H if SIDE = 'R'.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= (1,M).

    WORK

              WORK is REAL array, dimension
                          (N) if SIDE = 'L'
                          or (M) if SIDE = 'R'
              WORK is not referenced if H has order < 11.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 122 of file slarfx.f.

  subroutine slarfy (character UPLO, integer N, real, dimension( * ) V, integer INCV, real TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK)

SLARFY

Purpose:

     SLARFY applies an elementary reflector, or Householder matrix, H,
     to an n x n symmetric matrix C, from both the left and the right.
     H is represented in the form
        H = I - tau * v * v'
     where  tau  is a scalar and  v  is a vector.
     If  tau  is  zero, then  H  is taken to be the unit matrix.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix C is stored.
              = 'U':  Upper triangle
              = 'L':  Lower triangle

    N

              N is INTEGER
              The number of rows and columns of the matrix C.  N >= 0.

    V

              V is REAL array, dimension
                      (1 + (N-1)*abs(INCV))
              The vector v as described above.

    INCV

              INCV is INTEGER
              The increment between successive elements of v.  INCV must
              not be zero.

    TAU

              TAU is REAL
              The value tau as described above.

    C

              C is REAL array, dimension (LDC, N)
              On entry, the matrix C.
              On exit, C is overwritten by H * C * H'.

    LDC

              LDC is INTEGER
              The leading dimension of the array C.  LDC >= max( 1, N ).

    WORK

              WORK is REAL array, dimension (N)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 110 of file slarfy.f.

  subroutine slargv (integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y, integer INCY, real, dimension( * ) C, integer INCC)

SLARGV generates a vector of plane rotations with real cosines and real sines.

Purpose:

     SLARGV generates a vector of real plane rotations, determined by
     elements of the real vectors x and y. For i = 1,2,...,n
        (  c(i)  s(i) ) ( x(i) ) = ( a(i) )
        ( -s(i)  c(i) ) ( y(i) ) = (   0  )

Parameters
    N

              N is INTEGER
              The number of plane rotations to be generated.

    X

              X is REAL array,
                             dimension (1+(N-1)*INCX)
              On entry, the vector x.
              On exit, x(i) is overwritten by a(i), for i = 1,...,n.

    INCX

              INCX is INTEGER
              The increment between elements of X. INCX > 0.

    Y

              Y is REAL array,
                             dimension (1+(N-1)*INCY)
              On entry, the vector y.
              On exit, the sines of the plane rotations.

    INCY

              INCY is INTEGER
              The increment between elements of Y. INCY > 0.

    C

              C is REAL array, dimension (1+(N-1)*INCC)
              The cosines of the plane rotations.

    INCC

              INCC is INTEGER
              The increment between elements of C. INCC > 0.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 106 of file slargv.f.

  subroutine slarrv (integer N, real VL, real VU, real, dimension( * ) D, real, dimension( * ) L, real PIVMIN, integer, dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, real MINRGP, real RTOL1, real RTOL2, real, dimension( * ) W, real, dimension( * ) WERR, real, dimension( * ) WGAP, integer, dimension( * ) IBLOCK, integer, dimension( * ) INDEXW, real, dimension( * ) GERS, real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Purpose:

     SLARRV computes the eigenvectors of the tridiagonal matrix
     T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
     The input eigenvalues should have been computed by SLARRE.

Parameters
    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    VL

              VL is REAL
              Lower bound of the interval that contains the desired
              eigenvalues. VL < VU. Needed to compute gaps on the left or right
              end of the extremal eigenvalues in the desired RANGE.

    VU

              VU is REAL
              Upper bound of the interval that contains the desired
              eigenvalues. VL < VU.
              Note: VU is currently not used by this implementation of SLARRV, VU is
              passed to SLARRV because it could be used compute gaps on the right end
              of the extremal eigenvalues. However, with not much initial accuracy in
              LAMBDA and VU, the formula can lead to an overestimation of the right gap
              and thus to inadequately early RQI 'convergence'. This is currently
              prevented this by forcing a small right gap. And so it turns out that VU
              is currently not used by this implementation of SLARRV.

    D

              D is REAL array, dimension (N)
              On entry, the N diagonal elements of the diagonal matrix D.
              On exit, D may be overwritten.

    L

              L is REAL array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the unit
              bidiagonal matrix L are in elements 1 to N-1 of L
              (if the matrix is not split.) At the end of each block
              is stored the corresponding shift as given by SLARRE.
              On exit, L is overwritten.

    PIVMIN

              PIVMIN is REAL
              The minimum pivot allowed in the Sturm sequence.

    ISPLIT

              ISPLIT is INTEGER array, dimension (N)
              The splitting points, at which T breaks up into blocks.
              The first block consists of rows/columns 1 to
              ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
              through ISPLIT( 2 ), etc.

    M

              M is INTEGER
              The total number of input eigenvalues.  0 <= M <= N.

    DOL

              DOL is INTEGER

    DOU

              DOU is INTEGER
              If the user wants to compute only selected eigenvectors from all
              the eigenvalues supplied, he can specify an index range DOL:DOU.
              Or else the setting DOL=1, DOU=M should be applied.
              Note that DOL and DOU refer to the order in which the eigenvalues
              are stored in W.
              If the user wants to compute only selected eigenpairs, then
              the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
              computed eigenvectors. All other columns of Z are set to zero.

    MINRGP

              MINRGP is REAL

    RTOL1

              RTOL1 is REAL

    RTOL2

              RTOL2 is REAL
               Parameters for bisection.
               An interval [LEFT,RIGHT] has converged if
               RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

    W

              W is REAL array, dimension (N)
              The first M elements of W contain the APPROXIMATE eigenvalues for
              which eigenvectors are to be computed.  The eigenvalues
              should be grouped by split-off block and ordered from
              smallest to largest within the block ( The output array
              W from SLARRE is expected here ). Furthermore, they are with
              respect to the shift of the corresponding root representation
              for their block. On exit, W holds the eigenvalues of the
              UNshifted matrix.

    WERR

              WERR is REAL array, dimension (N)
              The first M elements contain the semiwidth of the uncertainty
              interval of the corresponding eigenvalue in W

    WGAP

              WGAP is REAL array, dimension (N)
              The separation from the right neighbor eigenvalue in W.

    IBLOCK

              IBLOCK is INTEGER array, dimension (N)
              The indices of the blocks (submatrices) associated with the
              corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
              W(i) belongs to the first block from the top, =2 if W(i)
              belongs to the second block, etc.

    INDEXW

              INDEXW is INTEGER array, dimension (N)
              The indices of the eigenvalues within each block (submatrix);
              for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
              i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.

    GERS

              GERS is REAL array, dimension (2*N)
              The N Gerschgorin intervals (the i-th Gerschgorin interval
              is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
              be computed from the original UNshifted matrix.

    Z

              Z is REAL array, dimension (LDZ, max(1,M) )
              If INFO = 0, the first M columns of Z contain the
              orthonormal eigenvectors of the matrix T
              corresponding to the input eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    ISUPPZ

              ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
              The support of the eigenvectors in Z, i.e., the indices
              indicating the nonzero elements in Z. The I-th eigenvector
              is nonzero only in elements ISUPPZ( 2*I-1 ) through
              ISUPPZ( 2*I ).

    WORK

              WORK is REAL array, dimension (12*N)

    IWORK

              IWORK is INTEGER array, dimension (7*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              > 0:  A problem occurred in SLARRV.
              < 0:  One of the called subroutines signaled an internal problem.
                    Needs inspection of the corresponding parameter IINFO
                    for further information.
              =-1:  Problem in SLARRB when refining a child's eigenvalues.
              =-2:  Problem in SLARRF when computing the RRR of a child.
                    When a child is inside a tight cluster, it can be difficult
                    to find an RRR. A partial remedy from the user's point of
                    view is to make the parameter MINRGP smaller and recompile.
                    However, as the orthogonality of the computed vectors is
                    proportional to 1/MINRGP, the user should be aware that
                    he might be trading in precision when he decreases MINRGP.
              =-3:  Problem in SLARRB when refining a single eigenvalue
                    after the Rayleigh correction was rejected.
              = 5:  The Rayleigh Quotient Iteration failed to converge to
                    full accuracy in MAXITR steps.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2016

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

Definition at line 294 of file slarrv.f.

  subroutine slartv (integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y, integer INCY, real, dimension( * ) C, real, dimension( * ) S, integer INCC)

SLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors.

Purpose:

     SLARTV applies a vector of real plane rotations to elements of the
     real vectors x and y. For i = 1,2,...,n
        ( x(i) ) := (  c(i)  s(i) ) ( x(i) )
        ( y(i) )    ( -s(i)  c(i) ) ( y(i) )

Parameters
    N

              N is INTEGER
              The number of plane rotations to be applied.

    X

              X is REAL array,
                             dimension (1+(N-1)*INCX)
              The vector x.

    INCX

              INCX is INTEGER
              The increment between elements of X. INCX > 0.

    Y

              Y is REAL array,
                             dimension (1+(N-1)*INCY)
              The vector y.

    INCY

              INCY is INTEGER
              The increment between elements of Y. INCY > 0.

    C

              C is REAL array, dimension (1+(N-1)*INCC)
              The cosines of the plane rotations.

    S

              S is REAL array, dimension (1+(N-1)*INCC)
              The sines of the plane rotations.

    INCC

              INCC is INTEGER
              The increment between elements of C and S. INCC > 0.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 110 of file slartv.f.

  subroutine slaswp (integer N, real, dimension( lda, * ) A, integer LDA, integer K1, integer K2, integer, dimension( * ) IPIV, integer INCX)

SLASWP performs a series of row interchanges on a general rectangular matrix.

Purpose:

     SLASWP performs a series of row interchanges on the matrix A.
     One row interchange is initiated for each of rows K1 through K2 of A.

Parameters
    N

              N is INTEGER
              The number of columns of the matrix A.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the matrix of column dimension N to which the row
              interchanges will be applied.
              On exit, the permuted matrix.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.

    K1

              K1 is INTEGER
              The first element of IPIV for which a row interchange will
              be done.

    K2

              K2 is INTEGER
              (K2-K1+1) is the number of elements of IPIV for which a row
              interchange will be done.

    IPIV

              IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
              The vector of pivot indices. Only the elements in positions
              K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
              IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
              interchanged.

    INCX

              INCX is INTEGER
              The increment between successive values of IPIV. If INCX
              is negative, the pivots are applied in reverse order.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2017

Further Details:

      Modified by
       R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Definition at line 117 of file slaswp.f.

  subroutine slatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)

SLATBS solves a triangular banded system of equations.

Purpose:

     SLATBS solves one of the triangular systems
        A *x = s*b  or  A**T*x = s*b
     with scaling to prevent overflow, where A is an upper or lower
     triangular band matrix.  Here A**T denotes the transpose of A, x and b
     are n-element vectors, and s is a scaling factor, usually less than
     or equal to 1, chosen so that the components of x will be less than
     the overflow threshold.  If the unscaled problem will not cause
     overflow, the Level 2 BLAS routine STBSV is called.  If the matrix A
     is singular (A(j,j) = 0 for some j), then s is set to 0 and a
     non-trivial solution to A*x = 0 is returned.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  Solve A * x = s*b  (No transpose)
              = 'T':  Solve A**T* x = s*b  (Transpose)
              = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    NORMIN

              NORMIN is CHARACTER*1
              Specifies whether CNORM has been set or not.
              = 'Y':  CNORM contains the column norms on entry
              = 'N':  CNORM is not set on entry.  On exit, the norms will
                      be computed and stored in CNORM.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of subdiagonals or superdiagonals in the
              triangular matrix A.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB,N)
              The upper or lower triangular band matrix A, stored in the
              first KD+1 rows of the array. The j-th column of A is stored
              in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    X

              X is REAL array, dimension (N)
              On entry, the right hand side b of the triangular system.
              On exit, X is overwritten by the solution vector x.

    SCALE

              SCALE is REAL
              The scaling factor s for the triangular system
                 A * x = s*b  or  A**T* x = s*b.
              If SCALE = 0, the matrix A is singular or badly scaled, and
              the vector x is an exact or approximate solution to A*x = 0.

    CNORM

              CNORM is REAL array, dimension (N)
              If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
              contains the norm of the off-diagonal part of the j-th column
              of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
              to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
              must be greater than or equal to the 1-norm.
              If NORMIN = 'N', CNORM is an output argument and CNORM(j)
              returns the 1-norm of the offdiagonal part of the j-th column
              of A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      A rough bound on x is computed; if that is less than overflow, STBSV
      is called, otherwise, specific code is used which checks for possible
      overflow or divide-by-zero at every operation.
      A columnwise scheme is used for solving A*x = b.  The basic algorithm
      if A is lower triangular is
           x[1:n] := b[1:n]
           for j = 1, ..., n
                x(j) := x(j) / A(j,j)
                x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
           end
      Define bounds on the components of x after j iterations of the loop:
         M(j) = bound on x[1:j]
         G(j) = bound on x[j+1:n]
      Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
      Then for iteration j+1 we have
         M(j+1) <= G(j) / | A(j+1,j+1) |
         G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
      where CNORM(j+1) is greater than or equal to the infinity-norm of
      column j+1 of A, not counting the diagonal.  Hence
         G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                      1<=i<=j
      and
         |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                       1<=i< j
      Since |x(j)| <= M(j), we use the Level 2 BLAS routine STBSV if the
      reciprocal of the largest M(j), j=1,..,n, is larger than
      max(underflow, 1/overflow).
      The bound on x(j) is also used to determine when a step in the
      columnwise method can be performed without fear of overflow.  If
      the computed bound is greater than a large constant, x is scaled to
      prevent overflow, but if the bound overflows, x is set to 0, x(j) to
      1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
      Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
      algorithm for A upper triangular is
           for j = 1, ..., n
                x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
           end
      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
           M(j) = bound on x(i), 1<=i<=j
      The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
      add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
      Then the bound on x(j) is
           M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
                <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                          1<=i<=j
      and we can safely call STBSV if 1/M(n) and 1/G(n) are both greater
      than max(underflow, 1/overflow).

Definition at line 244 of file slatbs.f.

  subroutine slatdf (integer IJOB, integer N, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV)

SLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.

Purpose:

     SLATDF uses the LU factorization of the n-by-n matrix Z computed by
     SGETC2 and computes a contribution to the reciprocal Dif-estimate
     by solving Z * x = b for x, and choosing the r.h.s. b such that
     the norm of x is as large as possible. On entry RHS = b holds the
     contribution from earlier solved sub-systems, and on return RHS = x.
     The factorization of Z returned by SGETC2 has the form Z = P*L*U*Q,
     where P and Q are permutation matrices. L is lower triangular with
     unit diagonal elements and U is upper triangular.

Parameters
    IJOB

              IJOB is INTEGER
              IJOB = 2: First compute an approximative null-vector e
                  of Z using SGECON, e is normalized and solve for
                  Zx = +-e - f with the sign giving the greater value
                  of 2-norm(x). About 5 times as expensive as Default.
              IJOB .ne. 2: Local look ahead strategy where all entries of
                  the r.h.s. b is chosen as either +1 or -1 (Default).

    N

              N is INTEGER
              The number of columns of the matrix Z.

    Z

              Z is REAL array, dimension (LDZ, N)
              On entry, the LU part of the factorization of the n-by-n
              matrix Z computed by SGETC2:  Z = P * L * U * Q

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDA >= max(1, N).

    RHS

              RHS is REAL array, dimension N.
              On entry, RHS contains contributions from other subsystems.
              On exit, RHS contains the solution of the subsystem with
              entries according to the value of IJOB (see above).

    RDSUM

              RDSUM is REAL
              On entry, the sum of squares of computed contributions to
              the Dif-estimate under computation by STGSYL, where the
              scaling factor RDSCAL (see below) has been factored out.
              On exit, the corresponding sum of squares updated with the
              contributions from the current sub-system.
              If TRANS = 'T' RDSUM is not touched.
              NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.

    RDSCAL

              RDSCAL is REAL
              On entry, scaling factor used to prevent overflow in RDSUM.
              On exit, RDSCAL is updated w.r.t. the current contributions
              in RDSUM.
              If TRANS = 'T', RDSCAL is not touched.
              NOTE: RDSCAL only makes sense when STGSY2 is called by
                    STGSYL.

    IPIV

              IPIV is INTEGER array, dimension (N).
              The pivot indices; for 1 <= i <= N, row i of the
              matrix has been interchanged with row IPIV(i).

    JPIV

              JPIV is INTEGER array, dimension (N).
              The pivot indices; for 1 <= j <= N, column j of the
              matrix has been interchanged with column JPIV(j).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    June 2016

Further Details:
    This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

      [1] Bo Kagstrom and Lars Westin,
          Generalized Schur Methods with Condition Estimators for
          Solving the Generalized Sylvester Equation, IEEE Transactions
          on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
      [2] Peter Poromaa,
          On Efficient and Robust Estimators for the Separation
          between two Regular Matrix Pairs with Applications in
          Condition Estimation. Report IMINF-95.05, Departement of
          Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.

Definition at line 173 of file slatdf.f.

  subroutine slatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, real, dimension( * ) AP, real, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)

SLATPS solves a triangular system of equations with the matrix held in packed storage.

Purpose:

     SLATPS solves one of the triangular systems
        A *x = s*b  or  A**T*x = s*b
     with scaling to prevent overflow, where A is an upper or lower
     triangular matrix stored in packed form.  Here A**T denotes the
     transpose of A, x and b are n-element vectors, and s is a scaling
     factor, usually less than or equal to 1, chosen so that the
     components of x will be less than the overflow threshold.  If the
     unscaled problem will not cause overflow, the Level 2 BLAS routine
     STPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
     then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  Solve A * x = s*b  (No transpose)
              = 'T':  Solve A**T* x = s*b  (Transpose)
              = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    NORMIN

              NORMIN is CHARACTER*1
              Specifies whether CNORM has been set or not.
              = 'Y':  CNORM contains the column norms on entry
              = 'N':  CNORM is not set on entry.  On exit, the norms will
                      be computed and stored in CNORM.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              The upper or lower triangular matrix A, packed columnwise in
              a linear array.  The j-th column of A is stored in the array
              AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    X

              X is REAL array, dimension (N)
              On entry, the right hand side b of the triangular system.
              On exit, X is overwritten by the solution vector x.

    SCALE

              SCALE is REAL
              The scaling factor s for the triangular system
                 A * x = s*b  or  A**T* x = s*b.
              If SCALE = 0, the matrix A is singular or badly scaled, and
              the vector x is an exact or approximate solution to A*x = 0.

    CNORM

              CNORM is REAL array, dimension (N)
              If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
              contains the norm of the off-diagonal part of the j-th column
              of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
              to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
              must be greater than or equal to the 1-norm.
              If NORMIN = 'N', CNORM is an output argument and CNORM(j)
              returns the 1-norm of the offdiagonal part of the j-th column
              of A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      A rough bound on x is computed; if that is less than overflow, STPSV
      is called, otherwise, specific code is used which checks for possible
      overflow or divide-by-zero at every operation.
      A columnwise scheme is used for solving A*x = b.  The basic algorithm
      if A is lower triangular is
           x[1:n] := b[1:n]
           for j = 1, ..., n
                x(j) := x(j) / A(j,j)
                x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
           end
      Define bounds on the components of x after j iterations of the loop:
         M(j) = bound on x[1:j]
         G(j) = bound on x[j+1:n]
      Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
      Then for iteration j+1 we have
         M(j+1) <= G(j) / | A(j+1,j+1) |
         G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
      where CNORM(j+1) is greater than or equal to the infinity-norm of
      column j+1 of A, not counting the diagonal.  Hence
         G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                      1<=i<=j
      and
         |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                       1<=i< j
      Since |x(j)| <= M(j), we use the Level 2 BLAS routine STPSV if the
      reciprocal of the largest M(j), j=1,..,n, is larger than
      max(underflow, 1/overflow).
      The bound on x(j) is also used to determine when a step in the
      columnwise method can be performed without fear of overflow.  If
      the computed bound is greater than a large constant, x is scaled to
      prevent overflow, but if the bound overflows, x is set to 0, x(j) to
      1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
      Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
      algorithm for A upper triangular is
           for j = 1, ..., n
                x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
           end
      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
           M(j) = bound on x(i), 1<=i<=j
      The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
      add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
      Then the bound on x(j) is
           M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
                <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                          1<=i<=j
      and we can safely call STPSV if 1/M(n) and 1/G(n) are both greater
      than max(underflow, 1/overflow).

Definition at line 231 of file slatps.f.

  subroutine slatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)

SLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

     SLATRS solves one of the triangular systems
        A *x = s*b  or  A**T*x = s*b
     with scaling to prevent overflow.  Here A is an upper or lower
     triangular matrix, A**T denotes the transpose of A, x and b are
     n-element vectors, and s is a scaling factor, usually less than
     or equal to 1, chosen so that the components of x will be less than
     the overflow threshold.  If the unscaled problem will not cause
     overflow, the Level 2 BLAS routine STRSV is called.  If the matrix A
     is singular (A(j,j) = 0 for some j), then s is set to 0 and a
     non-trivial solution to A*x = 0 is returned.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  Solve A * x = s*b  (No transpose)
              = 'T':  Solve A**T* x = s*b  (Transpose)
              = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    NORMIN

              NORMIN is CHARACTER*1
              Specifies whether CNORM has been set or not.
              = 'Y':  CNORM contains the column norms on entry
              = 'N':  CNORM is not set on entry.  On exit, the norms will
                      be computed and stored in CNORM.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading n by n
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading n by n lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max (1,N).

    X

              X is REAL array, dimension (N)
              On entry, the right hand side b of the triangular system.
              On exit, X is overwritten by the solution vector x.

    SCALE

              SCALE is REAL
              The scaling factor s for the triangular system
                 A * x = s*b  or  A**T* x = s*b.
              If SCALE = 0, the matrix A is singular or badly scaled, and
              the vector x is an exact or approximate solution to A*x = 0.

    CNORM

              CNORM is REAL array, dimension (N)
              If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
              contains the norm of the off-diagonal part of the j-th column
              of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
              to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
              must be greater than or equal to the 1-norm.
              If NORMIN = 'N', CNORM is an output argument and CNORM(j)
              returns the 1-norm of the offdiagonal part of the j-th column
              of A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      A rough bound on x is computed; if that is less than overflow, STRSV
      is called, otherwise, specific code is used which checks for possible
      overflow or divide-by-zero at every operation.
      A columnwise scheme is used for solving A*x = b.  The basic algorithm
      if A is lower triangular is
           x[1:n] := b[1:n]
           for j = 1, ..., n
                x(j) := x(j) / A(j,j)
                x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
           end
      Define bounds on the components of x after j iterations of the loop:
         M(j) = bound on x[1:j]
         G(j) = bound on x[j+1:n]
      Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
      Then for iteration j+1 we have
         M(j+1) <= G(j) / | A(j+1,j+1) |
         G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
      where CNORM(j+1) is greater than or equal to the infinity-norm of
      column j+1 of A, not counting the diagonal.  Hence
         G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                      1<=i<=j
      and
         |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                       1<=i< j
      Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV if the
      reciprocal of the largest M(j), j=1,..,n, is larger than
      max(underflow, 1/overflow).
      The bound on x(j) is also used to determine when a step in the
      columnwise method can be performed without fear of overflow.  If
      the computed bound is greater than a large constant, x is scaled to
      prevent overflow, but if the bound overflows, x is set to 0, x(j) to
      1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
      Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
      algorithm for A upper triangular is
           for j = 1, ..., n
                x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
           end
      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
           M(j) = bound on x(i), 1<=i<=j
      The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
      add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
      Then the bound on x(j) is
           M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
                <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                          1<=i<=j
      and we can safely call STRSV if 1/M(n) and 1/G(n) are both greater
      than max(underflow, 1/overflow).

Definition at line 240 of file slatrs.f.

  subroutine slauu2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)

SLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).

Purpose:

     SLAUU2 computes the product U * U**T or L**T * L, where the triangular
     factor U or L is stored in the upper or lower triangular part of
     the array A.
     If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
     overwriting the factor U in A.
     If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
     overwriting the factor L in A.
     This is the unblocked form of the algorithm, calling Level 2 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the triangular factor stored in the array A
              is upper or lower triangular:
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the triangular factor U or L.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the triangular factor U or L.
              On exit, if UPLO = 'U', the upper triangle of A is
              overwritten with the upper triangle of the product U * U**T;
              if UPLO = 'L', the lower triangle of A is overwritten with
              the lower triangle of the product L**T * L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 104 of file slauu2.f.

  subroutine slauum (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)

SLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).

Purpose:

     SLAUUM computes the product U * U**T or L**T * L, where the triangular
     factor U or L is stored in the upper or lower triangular part of
     the array A.
     If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
     overwriting the factor U in A.
     If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
     overwriting the factor L in A.
     This is the blocked form of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the triangular factor stored in the array A
              is upper or lower triangular:
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the triangular factor U or L.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the triangular factor U or L.
              On exit, if UPLO = 'U', the upper triangle of A is
              overwritten with the upper triangle of the product U * U**T;
              if UPLO = 'L', the lower triangle of A is overwritten with
              the lower triangle of the product L**T * L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 104 of file slauum.f.

  subroutine srscl (integer N, real SA, real, dimension( * ) SX, integer INCX)

SRSCL multiplies a vector by the reciprocal of a real scalar.

Purpose:

     SRSCL multiplies an n-element real vector x by the real scalar 1/a.
     This is done without overflow or underflow as long as
     the final result x/a does not overflow or underflow.

Parameters
    N

              N is INTEGER
              The number of components of the vector x.

    SA

              SA is REAL
              The scalar a which is used to divide each component of x.
              SA must be >= 0, or the subroutine will divide by zero.

    SX

              SX is REAL array, dimension
                             (1+(N-1)*abs(INCX))
              The n-element vector x.

    INCX

              INCX is INTEGER
              The increment between successive values of the vector SX.
              > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Definition at line 86 of file srscl.f.

  subroutine stprfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldwork, * ) WORK, integer LDWORK)

STPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.

Purpose:

     STPRFB applies a real "triangular-pentagonal" block reflector H or its
     conjugate transpose H^H to a real matrix C, which is composed of two
     blocks A and B, either from the left or right.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply H or H^H from the Left
              = 'R': apply H or H^H from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply H (No transpose)
              = 'C': apply H^H (Conjugate transpose)

    DIRECT

              DIRECT is CHARACTER*1
              Indicates how H is formed from a product of elementary
              reflectors
              = 'F': H = H(1) H(2) . . . H(k) (Forward)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Indicates how the vectors which define the elementary
              reflectors are stored:
              = 'C': Columns
              = 'R': Rows

    M

              M is INTEGER
              The number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B.
              N >= 0.

    K

              K is INTEGER
              The order of the matrix T, i.e. the number of elementary
              reflectors whose product defines the block reflector.
              K >= 0.

    L

              L is INTEGER
              The order of the trapezoidal part of V.
              K >= L >= 0.  See Further Details.

    V

              V is REAL array, dimension
                                    (LDV,K) if STOREV = 'C'
                                    (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                    (LDV,N) if STOREV = 'R' and SIDE = 'R'
              The pentagonal matrix V, which contains the elementary reflectors
              H(1), H(2), ..., H(K).  See Further Details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
              if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
              if STOREV = 'R', LDV >= K.

    T

              T is REAL array, dimension (LDT,K)
              The triangular K-by-K matrix T in the representation of the
              block reflector.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.
              LDT >= K.

    A

              A is REAL array, dimension
              (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'
              On entry, the K-by-N or M-by-K matrix A.
              On exit, A is overwritten by the corresponding block of
              H*C or H^H*C or C*H or C*H^H.  See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,K);
              If SIDE = 'R', LDA >= max(1,M).

    B

              B is REAL array, dimension (LDB,N)
              On entry, the M-by-N matrix B.
              On exit, B is overwritten by the corresponding block of
              H*C or H^H*C or C*H or C*H^H.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.
              LDB >= max(1,M).

    WORK

              WORK is REAL array, dimension
              (LDWORK,N) if SIDE = 'L',
              (LDWORK,K) if SIDE = 'R'.

    LDWORK

              LDWORK is INTEGER
              The leading dimension of the array WORK.
              If SIDE = 'L', LDWORK >= K;
              if SIDE = 'R', LDWORK >= M.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date
    December 2016

Further Details:

      The matrix C is a composite matrix formed from blocks A and B.
      The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
      and if SIDE = 'L', A is of size K-by-N.
      If SIDE = 'R' and DIRECT = 'F', C = [A B].
      If SIDE = 'L' and DIRECT = 'F', C = [A]
                                          [B].
      If SIDE = 'R' and DIRECT = 'B', C = [B A].
      If SIDE = 'L' and DIRECT = 'B', C = [B]
                                          [A].
      The pentagonal matrix V is composed of a rectangular block V1 and a
      trapezoidal block V2.  The size of the trapezoidal block is determined by
      the parameter L, where 0<=L<=K.  If L=K, the V2 block of V is triangular;
      if L=0, there is no trapezoidal block, thus V = V1 is rectangular.
      If DIRECT = 'F' and STOREV = 'C':  V = [V1]
                                             [V2]
         - V2 is upper trapezoidal (first L rows of K-by-K upper triangular)
      If DIRECT = 'F' and STOREV = 'R':  V = [V1 V2]
         - V2 is lower trapezoidal (first L columns of K-by-K lower triangular)
      If DIRECT = 'B' and STOREV = 'C':  V = [V2]
                                             [V1]
         - V2 is lower trapezoidal (last L rows of K-by-K lower triangular)
      If DIRECT = 'B' and STOREV = 'R':  V = [V2 V1]
         - V2 is upper trapezoidal (last L columns of K-by-K upper triangular)
      If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.
      If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.
      If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.
      If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.

Definition at line 253 of file stprfb.f.

Author

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